LOGIC COLLOQUIUM '82 Proceedings of the Colloquium held in Florence 2328 August, 1982
Edited by
G. LOLL1 Dipartimento di lnformatica Univer.sitadi Torino Torino Italy
G. LONG0 Dipartimenio di lnformatica Universita di Pisa
Pisn Itnly
and
A. MARCJA Dipartimento di Matematica Lihern Universita degli Stud di Trento Trento Italy
1984
NORTHHOLLAN'D AMSTERDAM 0 NEW YORK. OXFORD
QELSEVIER SCIENCE PUBLISHERS B . V . , 1984 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
ISBN: 0 444 86876 3
Published by: Elsevier Science Publishers B . V P.O. Box 1991 1000 BZ Amsterdam The Netherlands Sole distributors for the U . S .A . and Canada. Elsevier Science Publishing Company, Inq; 52 Vanderbilt Avenue New York, N . Y . 10017 U.S.A.
Library of Congress Cataloging in Publication Data
Logic Colloquium (1982 : Florence, Italy) Logic colloquium '82. (Studies in logic and the foundations of mathematics ; v. 112)
Bibliography: p. 1. Model theoryCongresses. matics)Congresses. 3. Lambda I. Lolli, Gabriele, 1942111. Marcja, A. (Annalisa) IV.
.
@9*7*L64 1982 511' .8 ISBN 0444868763 ( U . S . )
2. Categories (MathecalculusCongresses. 11. Longo, Giuseppe. Title. V. Series.
841630
PRINTED IN T H E NETHERLANDS
V
PREFACE
The Logic Colloquium ‘82 has been held in Florence (Italy), from 23 to 28 August, 1982. The date had been so chosen in order to allow a combined participation to the Warsaw ICM. Notwithstanding the postponement of the latter, more than 200 logicians were able to attend the Logic Colloquium and the Summer Meeting of the ASL. The organization and the program of the conference were undertaken by a Committee consisting of E. Casari (Florence), J.E. Fenstad (Oslo), G . Lou (Turin), G . Longo (Pisa), A. Marcja (Trento), and D. van Dalen (Utrecht). The organization has been made possible by the financial support of many institutions: a substantial help came through the Comitato per la Matematica of the Italian Consiglio Nazionale delle Ricerche; further contributions were made accessible from the University of Florence, the city of Florence and the Regione Toscana, and the Ente Provinciale per il Turismo di Firenze. The conference was sponsored by the International Union of History and Philosophy of Science, Division of Logic, Methodology and Philosophy of Science. The main topics chosen for the Colloquium were: Model Theory, with particular emphasis on models of Peano arithmetic (a small if not exactly pertinent contribution to the celebrations for the 50th anniversary of Peano’s death); Categorical Logic; Lambdacalculus. Much care and effort have been put by the invited speakers into investigating and further stretching the overlapping and crossbreeding of these areas. We regret that the present volume does not faithfully reflect the balance of the three topics at the Logic Colloquium, since some of the speakers in categorical logic and in lambdacalculus were unable to submit the written text of their lectures (while the editors were guilty against the readers of too much waiting). We hope that their contributions will however be made accessible in other ways to all interested logicians. We are grateful to Elsevier Science Publishers B.V. for inserting this volume in their glorious series in Logic and the Foundations of Mathematics. November 1983
G . Lolli (Torino) G . Longo (Pisa) A. Marja (Trento)
LOGIC COLLOQUIUM '82 G. Lolli, G.Long0 and A. Marqa (editors) @Elsevier Science Publishers B. V. (NorthHolland), 1984
1
LECTURES ON NONSTANDARD MODELS OF ARITHMETIC
Commemorating Guiseppe Peano C. Smoryikki Department of Mathematics The Ohio State University Columbus, OH 43210
USA
Contents § 0.
1. 2. 3.
4. 5. 6. 7. 8. 9. 10.
Preface The Beginnings (The 1950s and Earlier) The R6le of the Infinite Integer (The 1960s. I) Extensions of Models (The 196Os, 11) Saturation Properties (The 1970s. I) Recursively Saturated Models of Rich Theories (A Digression) The Arithmetised Completeness Theorem (The 197Os, 11) Powerful Arithmetisations (The 1970s, 111) Diversity (The 1980s) A Dead End Summer Reading List
0. PREFACE Guiseppe Peano died in 1932; nonstandard models of arithmetic were born the following year. Thus, Peano never studied models of arithmetic and it may seem odd to commemorate him with a series of lectures on such a topic. However, one of the more active areas in the study of what has come to be called Peano arithmetic, o r P A , is the study of its nonstandard models and this subject seems thus appropriate. Besides, with the subject beginning just after his death, we might say it picked up where he left off. It is my intention subject. so.
to
give a partly historical account of the development of the
Since these are commemorative lectures, it seems most appropriate to do
Moreover, there are now two collections of papers edited by Kenneth McAloon
e t aZ. with expositions of recent work, while there is nothing about the pioneering efforts of the 1950s and 1960s.
Progress was much greater in this period
than most people realise.
I will depart from the historical line in two important respects.
First, much
early work concerned only "strong" nonstandard models of arithmetic rather than arbitrary models of P A .
A nonstandard model was called "strong" if it elemen
tarily extended the standard model No= (w;+,.,',O)
and was of particular interest
only because of a lack of familiarity with the strength of P A :
One needed to
know, say, that all integers (standard and nonstandard) had certain coding prop
c. S M O R ~ S K I erties that ordinary integers had and, unaware that such properties were provable outright in P A , one simply assumed them to hold by assuming the model to be strong. A second departure from the historical line must occur when several independent trends develop simultaneously. This first happened around 1960, when nonstandard model theory reached its first level of maturity. Because of time and space limitations, I shall have to assume the reader familiar with a great deal of what can be done within PA.
This means that, with respect to,
say, the hypothesis of strength mentioned above, I can discuss the full results available without the irrelevant hypothesis even though the full result came a decade later.
It also means that some important developments, which preceded
formalisation, will look more trivial than they should. My notation will be fairly standard at least by my standards and I hope everything will be fairly selfexplanatory. 1.
THE BEGINNINGS (THE 1950s AND EARLIER)
It was in a paper published in 1929 that Thoralf Skolem first suggested the possibility that the standard numbers might not be alone, that there might be poor In SkoZem 1 9 3 3 , he was able to
imitations satisfying the same firstorder laws.
psove this relative to any finite set of axioms. This, of course, could have been established by appeal to Gb'del's Incompleteness and Completeness Theorems; or, it could have been done by the nowcommon appeal to the Compactness Theorem.
The
former approach would seem to have little bearing on the larger problem of obtaining nonstandard models satisfying a l l the true sentences of arithmetic; and neither of these abstract approaches would very likely have been to Skolem's taste. In any event, in the following year in SkoZem 1934 he published a proof of the existence of strong nonstandard models of arithmetic, i.e. structures 1'4 =
(M;+,,',O)
not isomorphic to No but, nonetheless, making true the same sentences
as N o . Skolem's original papers were in German.
He republished the proof in English in
SkoZem 1955 in the proceedings of a conference the same proceedings in which bos published his theorem on ultraproducts. Although this was merely a repetition
of his earlier proof, it is this later paper to which most people refer.
Perhaps
this is because this paper is in English, a much easier language than German for many of us; perhaps it is simply that this paper, appearing in a slim little volume that has been reprinted, is in more private libraries and hence more accessible; or perhaps it is the juxtaposition with bos' paper that strikes one's fancy for, Skolem's construction of a nonstandard model is something of an ultrapower construction.
Sketch of Skolem's construction:
Given No
=
(u;+,*,',O),
let F consist of
3
Lectures on Nonstandard Models of Arithmetic all functions F:w
+
w
definable in No.
... of F it is
From an enumeration FO,F1,
not difficult to construct, by diagonalisation, a function G:w all F ,F. i J
E
+
w
such that, for
F,
< F.G(x) eventually F G(x) i J or eventually F.G(x) > F . G ( r ) .
or
eventually F.G(x) = F.G(x) J
J
Using G, one can define an equivalence relation :by F. 1
5
F. iff eventually F.G(x) = F.G(r) J
J
and shew the structure F/: to be an elementary extension of N 0 '
It should be noted that, although Skolem's construction resembles the ultrapower construction so much that one feels like calling it such, an important element is missing.
Skolem's use of the diagonalising G in place of the nowusual ultrafil
ter relies heavily on the countability of the arithmetic language; his method does not yield the existence of nonstandard models when, say, a continuum of predicates naming all sets of natural numbers is added to the language. For this latter, one must use one of the standard abstract existence theorems of logic. It is also worth mentioning that Skolem's goal in constructing nonstandard models was philosophical: He aimed to shew that firstorder logic could not characterise the number series; he did not care to start a new subject.
Until the 1960s, this
was generally the case nonstandard models of arithmetic were either objects of philosophical interest or tools, not objects of mathematical interest in their own right.
The major counterexample to this was an observation made by Leon Hen
kin in his paper Henkin 1950 on the Completeness Theorem for Type Theory. announced the order type of a nonstandard model of arithmetic to be w where 8 is a dense linear order.
( Exercise: If, in particular,
+
He
(w*+u)8,
0 is countable,
it must be the order type of the rationals. A related, but trickier, exercise due (I believe) to Klaus Potthoff is this: not have the order type of the reals.
Shew, in the uncountable case, 0 can
)
The important, i.e. useful, fact about the order type of a model M = (M;+,*,*,O) of PA is that M begins with w and then follows this with the nonstandard or infi
nite integers. The significance of the infinite size of the nonstandard integers is manifold: They code various paths to infinity and, particularly in Nonstandard Analysis, simulste limit processes; they code infinite sets; and they code nonprincipal ultrafilters in the algebra of definable sets of natural numbers. all this, their potential usefulness is clear.
With
In the early 1950s, however, not
all of this was known: The first application of infinite integers depended merely on their size. In RyZZNardzewski 1952Czesaaw RyllNardzewski essentially proved the following theorem: 1.1.
Theorem.
PA is not finitely axiomatisable.
c. S M O R ~ S K I
4
RyllNardzewski actually proved something a bit stronger: If T is any finitely axiomatised theory in a language extending that of PA and T is true in some expansion of No to accommodate the extended language, then T does not prove some instance of induction in this language. This result is still rather weak.
As I
mentioned in the Preface, many early results were proven in weak form because it was necessary to assume certain arithmetic truths held in a nonstandard model and, through lack of familiarity with the power of PA, the best guarantee of this assumption was that the nonstandard model was a strong one.
Dropping this assump
tion, the final result along these lines is the following: 1.2.
Theorem.
PA is essentially unbounded, i.e. no consistent extension T of PA
can be given by axioms of a fixed bounded complexity. This theorem was first published by Michael Rabin in Rubin 1962 and is thus often referred to as Rabin's Theorem, although he explicitly announced it to have been known to others most probably including Solomon Feferman, Georg Kreisel, Dana Scott, Stanley Tennenbaum, and Hao Wang.
A s the reader might guess from the long
list of names, the road from Theorem 1.1 to Theorem 1.2 was a long one.
Indeed,
it cap now be recognised to have been one of the two major themes in nonstandard model theory in the 1950s, if one may refer to the thenprimitive development as "nonstandard model theory".
Today, we have many proofs of Theorems 1.1 and 1.2,
among which is a modification of RyllNardzewski's proof of Theorem 1.1. Let M be an arbitrary model of T ? P A and let MI be an elem
Proof of 1.2:
0
entary extension of M which contains at least one nonstandard integer a which is 0
infinite relative to M ~ i.e. , a > b for every element b
E
J M ~ J .Suppose n is a
finite positive integer and U is a set of C sentences true in Mo. struct a model Mz
b
U but such that M2
y
The construction is very simple: Let IN
We will con
PA. 2
I
be the closure of IN
I
{ a } under all M 2 is automatic
0 "
total functions definable in Ml by parameterfree Znformulae. ally a Cnelementary substructure of M I , whence it satisfies U.
The reason that M 2 is not a model of PA is also simple: There is a Cn+Idefinable function F which can schematically be proven in PA to eventually dominate each C definable function, i.e. for each parameterfree C formula natural number
I such
@VoVl

there is a
that
where Ji defines F. ( F is obtained by a simple diagonalisation on the uniform C enumeration of Zndefinable partial functions. ) In Mz, F is not total; in fact,

Fa is not defined.
To establish this last claim, suppose Fa exists, i.e. M 2 k 3 V 1V 2$'av102, where I )' is Il
and 3v2$'defines F .
Let b be the image of a under F in M2 and let c witness
5
Lectures on Nonstandard Models of Arithmetic this fact, i.e. M2 Fa = b in MI.
I=
$ 'abc.
function G such that b
=
Since $ ' E
M2
k $'&
E
calling that a > d for every d E lMo 1 , function H.
nn,
1M21, there is some d G(a,d) in Ml. Letting Hvo
Now, since b
=> Ml
IW 0 I
E
k I)'&?
, whence
and some C definable
supv HV ) and a is quite infinite, whence b
> :(FVO
0
=
Fa > Ha in
&ED I wish to emphasise here the r61e played by the size of a relative to Mo: The integer a is so large that a Zn+lfunction eventualzy dominating all C functions already dominates them at a.
Thus, the mere closure under Z functions, which
yields a C elementary substructure, does not yield closure under the provably total majorising function.
Through such an application, the mere size of an in
finite integer presents itself as a useful tool, one which could still be exploited unaided in the 1950s in the work of Feferman, Scott, and Tennenbaum, and even as recently as 1975 in a paper of Alex Wilkie. As I remarked just before proving Theorem 1.2, the passage from Theorem 1.1 to Theorem 1.2 was one of two main themes in the study of nonstandard models.
The
other, which nowadays one can see to partially merge with this one in the work of Feferman, Scott, and Tennenbaum, is a couple of decades older and more technical. This theme is the complexity of models and goes back to the second volume of Hilbert and Bernays, specifically to a result Paul Bernays proved and included in the volume: 1.3.
Theorem.
Let
0 be
any sentence of a given language.
If $I is consistent,
i.e. if $I has a model, then $I has an arithmetically definable model, i.e. a model whose domain is an arithmetically definable set of natural numbers and whose primitive relations are also arithmetically definable. Actually, the entire satisfaction relation for the model can be taken to be arithmetically definable the satisfaction relation for the language of @ is simply a new predicate describable by a new axiom @ and one can apply the result to $ A $ . The fact that the description could be given by a single axiom @ was not immediately recognised.
This was the decade* in which Stephen Kleene (KZeene 1952.4)
(and later William Craig and Robert Vaught) proved one could finitely axiomatise r.e. theories by the addition of new predicates.
Thus, when the early researchers
became interested in extending Bernays' result to r.e. theories, the direct reduction to the finite case by finitising the theory in telrms of a satisfaction relation did not occur to them and they proved the result anew. Bernays had proved Theorem 1.3 by arithmetically analysing the proof o f Godel's
*In fact, Robinson's finitely axiomatised Q was not published until 1953.
c. S M O R Y ~ K I
6
Completeness Theorem for the predicate calculus.
The new result was established
(by Gisbert Hasenjsger, Stephen Kleene, and Hao Wang) by doing the same to modifications of Henkin's proof. We will discuss this later in 96, where we will see that this final completion of Bernays' Theorem is a useful tool. This realisation came, however, around 1970; in the 1950s the question was simply one of the complexity of models. The upper bound on the complexity of a model of an r.e. theory is readily established by inspection: If T is a consistent r.e. theory, then the arithmetic encoding of the proof of the Completeness Theorem shews T to have a model on an initial segment of the set of natural numbers and the definition of satisfaction for this model (and consequently each of its primitive relations) to be A2.
[ Recursion theoretically, this means the model is recursive in

0'.
Later
improvements were made by Joseph Shoenfield and, ultimately, by Carl Jockusch and Robert Soare; cf. ShoenfieZd 1960 and Jockusch and Soare 2972A8B. ) For lower bounds on the complexities of the interpretations, one can again ask two questions: How complicated must the primitive relations be; and, how complicated must the satisfaction relation be? Well, one r.e. theory's satisfaction relation is another finitely axiomatised theory's primitive relation and, globally, the questions conflate.
For specific theories (like P A ) or fixed languages (e.g.
that of one binary relation symbol), however, the equivalence is nontrivial and one usually asks for large laver bounds on the complexities of the primitive relations. In the 1950s, Kreisel and Andrzej Mostowski alternated in a series of papers, the
main outcomes of which were the existence of r.e. and finitely axiomatised theories with no recursive models.
Mostowski even shewed that, whereas the set
of sentences true in all models is r.e., that of all sentences true in all recur
sive models is not even arithmetically definable. In 1960, Vaught offered yet another generalisation. In the late 1950s, the problem of the difficulty of constructing nonstandard models of PA, i.e. the complexity problem for models of P A , was considered by Feferman, Scott, and Tennenbaum, who published a short series of abstracts. Among other things, they reconsidered Skolem's ultrapowerlike construction and shewed, e.g., that no homomorphic image of the Xnfunctions could model PA.
This was
largely an application of size, 6 Za RyllNardzewski (or, at least, my exposition of the latte&
proof, above).
However, their work led Tennenbaum to an important
discovery: Infinite integers not only have infinite size, but they also code infinite sets.
Looking at the encoding yields:
1.4. Theorem (Tennenbaum's Theorem). Let M = (w;+,X,',O) be a nonstandard model of PA and let 4m be any formula of the language of arithmetic (with parameters
Lectures on Nonstandard Models of Arithmetic from M allowed).
X
=
Then In fact, X is recursive in each of
is recursive in i , x .
( Notes: i. Note that
M k G }
{ x E w :
+,x.
denotes the successor function of the model; 0 the zero element. is recursive in
isince X I
=
x
iI
, where 1 is the unit of M.
Without
loss of generality, we can assume 0 to coincide with 0 and use the simpler notation. and x,was not
In the first published proof in Ehrenfeuckt and KreiseZ
1966 it is noted that X is recursive in
credits the observation on
+.
to Feferman.
X
Kenneth McAloon, in M c A l o o n 1 9 8 2 ,
)
First, we recall that an integer x can serve as a code for a
Proof of 1 . 4 :
finite set of integers.
There are many ways in which this can be done, my least For an integer X, let Dz
favourite being one expecially suited for this proof. consist of all 9 ' s such that the (y +l)th where p o , p l ,
+
The extra flourish, that X is recursive in each of
ii.
stated by Tennenbaum.
prime divides x :
... is the sequence 2 , 3 , ... of primes.
y
E
iff p (z,
Dx
Y
A simple induction on v 0 proves the Aussonderungsaxiom, PA
1 vVo3VIv
V2(
V 2 E
Du 1
E V2
[email protected] ),
for any formula @ v 2 . Let a be an infinite integer in M, let $ be given, and let E w: M 1. Applying Aussonderung to @ and a, we obtain b E lMl such
6
X = { x
that X = { x
E
w: M
13: E
iff
This shews X to be r.e. in shews o

X to be r.e. in
Recursivity in
X
3 1.
But then
M I G I iff E 3 c E w ( ~ k rX * c = E )  3 , E w ( M k C i... ic b ) ( p , times).
iff
EX
f.
Since w  X corresponds to
i , whence
X is recursive in
is a similar affair:
the same proof
[email protected],
f.
From the equivalence between z E X and the
representability of b as a p fold sum of c's, we get the equivalence with the X
representability of Zb as a p fold product: x X b some d , where e = 2
E
X iff d
.
x
...
x
d
e , for
&ED
Tennenbaum's Theorem has some immediate corollaries:
1.5. Then
Corollary. +,X
Let M = (w;i,x,',O)
be a strong nonstandard model of arithmetic.
are not arithmetical.
This can, of course, be hierarchically refined. 1.6.
Corollary.
If M = ( w ; f , ~ , ~ , O )
The main such refinement is:
is a nonstandard model of P A , then +,x are
not recursive.
Proof:
It suffices to choose @ so that
This is a simple recursion theoretic trick:
x
E
o: M k
61
is not recursive.
If $,J, define two provably disjoint
effectively inseparable r.e. sets, say A,B, respectively, then A 5 X =
c. S M O R ~ S K I
8 { J:
E
61
o: M
and B n X = @, whence X is not recursive.
I have one more important corollary to cite.
BED
Before giving this, however, I must
remark that the proof of Corollary 1.6 did not depend on the full power of Tennenbaum's Theorem. fic formula
It really only depended on establishing Theorem 1.4 for one speci
@.For this, the proof needs:
i. some minimal arithmetic say,
Robinson's Q; ii. the totalities of the exponential and prime enumerating functions; and iii. two instances of induction to establish Aussonderung for
0,I$.
If we
conjoin these to obtain a single axiom 9, Corollary 1.6 can be rewritten as
1.6'.
Corollary. If M
= (w;+,x,',O)
b
$ is nonstandard, then
+,X
are not recur
sive. Conjoining with ii, any axiom forcing the model to be nonstandard yields a sentence having no recursive models.
In fact, with ii, we can easily prove Mostowski's
result:
1.7.
Corollary. The set of sentences valid in all recursive models is not
arithmetical.
Proof:
The only recursive model of $ is the standard one.
Hence, if fJ is any
arithmetic sentence, $ 4 0 is valid in all recursive models iff 0 is true in the standard one.
Since arithmetical truth is not arithmetical, the Corollary follows. QED
The sentence ii, consisting mainly of two odd instances of induction conjoined with a few natural axioms is a bit odd and one can ask for an aesthetically pleasing finite theory to which Corollary 1.6 can be applied.
It is not hard to see that
(over &) Z1 induction suffices to establish the totalities of the exponential and prime enumerating functions and C

1
and Il Aussonderung. 1
One can also ask for the weakest theory which can be used in place of $.
Z 1nduc1 tion is rather strong and one can use weaker theories, as shewn in McAZoon 1982.
Tennenbaum's Theorem can also be generalised in another direction, as we shall see in 55, below. I might also add that the question of the complexity of a model has of late resumed an interest; cf., e.g., Marker 1982. A
2.
THE ROLE OF THE INFINITE INTEGER (THE 1960s I)
The year 1960 was something of a watershed in the development of nonstandard model theory. Prior to 1960 most papers on nonstandard models of arithmetic were aimed at metamathematical targets, the nonstandard models themselves not being of central interest.
Then, suddenly, this changed: In the early part of the decade any num
ber of papers appeared in which the models themselves were studied. There were papers on the additive group of integers (negative ones added) of a nonstandard
9
Lectures on Nonstandard Models of Arithmetic model, on the nonstandard model as a semiring, on the algebra of definable sets in a nonstandard model, and on the forms of embeddability on nonstandard models into one another. The variety is such that a straight historical discussion of the period is pointless.
Instead, I shall outline a few central themes.
First, let me hark back to the r8les played by the infinite integers. While mere size considerations remained important (cf. for example Ehrenfeucht and KreiseZ 1 9 6 6 ) , greater sophistication arose.
The following theorem relating the infinite
to the finite integers is a cornerstone of Abraham Robinson's Nonstandard Analysis; in the proper language, it is the lemma upon which the equivalence of the standard and nonstandard definitions of the limits of sequences depends.
2.1.
Theorem. Let M be a strong nonstandard model of arithmetic, i.e. a proper
elementary extension of No. i. ii.
Proof: v o @vl
> vo $vl.
Because i and ii are duals, I shall only prove i. Observe: No
b
V V o 3v1 > v 0 @vl =>
M
k
Vv0 3Vl >
Vo
@v1. Picking u
infinite, the righthandside of this implication produces an infinite v
=>.
Let a be infinite such that M
MI=
a > ;*&
>
M1=3 v
Since cc was arbitrary, we conclude N 0
>
b 6.Let x E o and 3: @vl => No b 3v1
0
1'
observe
>
3: @ v l .
V V 0 3v1 > V 0 @ v l *
QED
This theorem is slightly disappointing in that it assumes M to eZementariZy extend
No.
There are hierarchical variants which the reader can work out for himself and
also something called Overspill to handle the nonelementary case.
I shall shortly
discuss the latter; but, before doing s o , I would like to remark that the present theorem, however much it assumes, is rather useful.
It is, for example, the key
to one of the corollaries of the main theorem of Jeff Paris and Leo Harrington: Paris and Harrington proved the independence of various true 112 sentences @ = V V 0 3 v 1 ~ 0 V 1 , with ii, E
the least y such that No tions.
A,.
The (recursive) function call it G taking x to
I= wT
eventually dominates all provably recursive func
These two facts are equivalent, the latter trivially implying the former.
One can, with some effort, prove directly the majorisation result and thereby conclude the underivability of $; conversely, and more easily, one can appeal to Theorem 2.1.11 and the model theoretic construction underlying the independence proof:
Most expositions of the ParisHarrington Theorem demonstrate the underiv
ability of $ by starting with an elementary extension M of No in which @ is true. For each infinite integer a
E
/MI and its image b = Ga, they shew the existence of
an initial segment I 5 IWI such that i. a the inherited functions. Since
E
I and b t! I,and ii. I I= PA when given
I is a model of
P A , it is closed under all provably
recursive functions. For such a function F, from the facts that Fa
E
I,Ga t! I,
c. S M O R Y ~ K I
10
and I is an initial segment of M, it follows that Fa < Ga. Theorem 2.l.ii yields: N o
b
Bvo vvl > v o (FVl >
Since a was arbitrary,
GOl).
I have omitted a few things here, such as an absoluteness of the provably recursive functions, and I have used much stronger conditions on M than are necessary for the above argument; but I have illustrated the use of the Theorem. The reader familiar with Ramsey’s Theorem may wish to use the other half of Theorem 2.1 to
( Similarly, the reader familiar
reduce the finite form to the infinite form.
with the work of Paris and Harrington may wish
to
reduce their variant to the
infinite Ramsey Theorem. The lazy reader may wish to wait for 56, below.
)
In the nonelementary case there is, as I said, a replacement for Theorem 2.1: 2.2.
(Weak Overspill).
v
5
0 1
Let
$v0V2
( M I= & )
iff
(Strong Overspill).
Let
x ii. $71
Let M
Theorem (Overspill Principle). i.
E
w
defines a function F : I M ( Vz
w
E
I= PA
have only
be nonstandard and b
Vo,VI
3 infinite a ( M @v0VIv2
+
E
(MI.
free. Then
I=
vv0
a , i.e. Fa is infinite.
&ED
The Overspill Principle is, in its weak form, apparently due independently to Rabin and Abraham Robinson and, in its strong form, to Robinson.
Weak Overspill
is used a great deal in the study of nonstandard models of arithmetic, as we shall
Lectures on Nonstandard Models of Arithmetic
11
shortly see.
For now, however, a simple application will suffice: Suppose a E 1 1 1
is infinite.
Then there is an infinite b
finite, so is ZX; whence 2" < a .
E
IMI such that 2b < a .
For, if x is
By Overspill there is also an infinite b such
b < a.
that 2
In its strong form, Overspill is extremely useful in Nonstandard Analysis.
Applied
in that language, with F replaced by l/F, it becomes the Infinitesimal Prolongation Theorem:
... is an internal sequence of infinitesimals, then for a few
If r o , r l ,
infinite integers a , r
remains infinitesimal. One can consult, e.g. Martin Davis'
book for examples of the usefulness of this principle.
In the study of nonstandard
models of arithmetic, however, Strong Overspill has not made much of an appearance until recently, and then only tangentially as an insight behind the Indicator Theory of Laurie Kirby and Jeff Paris.
Even here, unless there is some recent
work I am unaware o f , there are no results which cannot as easily be proven by other means.
( Hence,
I shall not discuss indicators in the sequel. )
Both Overspill and the preceding Theorem 2.1 are manifestations of the r81e of an infinite integer as a path to infinity: An infinite integer behaves somehow like the integers it encounters in its path.
In an ultrapower, an infinite integer is
literally such a path; in the more general case, my words are mystical but of some heuristic value.
In any event, I view these last two theorems as sophisticated
views of the sizes of infinite integers. in $ 1 was their coding power.
The other property of integers we used
Tennenbaum's Theorem was a crude mixture of this
coding power and size. We now take a more sophisticated look at this. 2.4.
Definitions. Let M be a model of arithmetic. A set X c w is standard on M
if there are a formula @ J ~ with u ~ only u O , V l free and an element b
E IMI such that x E w: M I= I@ 1. The collection of standard sets of M is called the standard system of M, written SSyIMl. A number a E IMI codes X E SSy(MI iff X =
X
=
I
.c E w:
MI=;
E
D;; 3 .
Here, by Dx I mean the finite set canonically indexed by
X.
We already used the
prime decomposition of numbers as a means of encoding finite sets.
Generally,
however, it is more convenient (i.e. I prefer) to use the following: Do
=
{
1;
D
=
,..., xn I
x0
1,
+ ... + ZXnI ... > xnI'
if x = 2"O
x* >
and
Before discussing standard sets and their codes, I should warn the reader that the terminology presented here is not universally used.
Following the deplorable
logical tradition of referring to sets of natural numbers as real numbers, the standard sets are often called the reazs of the model, SSyfMI being thus denoted
RM.
Moreover, for reasons soon to be evident, SSyfMI = R
Scott set.
M is often called a
(Well, actually SSy(MI is proven to be a Scott set and then called
the Scott set of the model.
)
c. S M O R Y ~ S K I
12
Tennenbaum's Theorem depended on the fact that in a nonstandard model every standard set possessed a code.
2.5.
A bit more is true:
Let M I = PA be nonstandard, X
Lemma.
E
SSyfMl.
X has arbitrarily small
infinite codes.
Proof:
Let X be defined through $vv,F, a an arbitrary infinite integer, c
infinite such that 2c < a , and F defined by FV
=
Then
{
1,
@vF
0,
+vF.
C2
d
c
=
~v
+ zC'
< 2'
iii follows from
We have already proven iii => i.
the existence of a minimum model M
T
to prove i => ii.
of T: SSy(MT) = R e p ( T ) .
Thus, it suffices
For the sake of convenience, I will only shew i=> iii. This
requires a lemma:
2.9.
Lemma.
Let
x be cclosed and let X
E
sistent theory T (in a recursive language). is also coded in
x.
x code a set of axioms for some conThen some consistent completion of P
roof: Let, for some axiomatisation of T , E
x.
x= I
'$':
is an axiom of T
A l s o , let I $ ~ , @ ~be , .a. .recursive enumeration of all sentences in the lan
guage of T .
A completion of
T is essentially an infinite path through the binary
tree
more exactly, it is an infinite such path consistent with 2'. is Ill in T, not recursive in T, and the fact that T
E
Consistency with P
x does n o t
guarantee
x to
c. S M O R Y ~ ~ S K I
14
include the tree of paths consistent with T .
eo
el
;!:$I,..., those paths $o ',I$]
However, it does include the set of
which have no proofs of their inconsistency with T
.~
of code at most k .
x+
=
I
/x\
where I) =
'
yclhkc) predicate calculus of The
+ set X
V].
+
coding an infinite path
+ through X .
E
Let T' have axioms I) for x X
leave to the reader to shew, a consistent, complete theory. it is recursive in its axiomatisation and T' = Before returning to the proof of Theorem 2.8,
'$':
x
x.
Choose Y f Y. T' is, as I Since T' is complete,
is an infinite binary tree recursive in X, whence X
T'b $
1
E
E
x.
&ED
let me pause to remark that this
lemma not only explains the terminology adopted, but it also provides a very Not only will we use it immediately,
useful property of completionclosed algebras. bur also we will apply it later.
Proof vf 2.8, continued:
x be a countable cclosed family of sets of ... . We construct a modelM with
Let
natural numbers with an enumeration Xo,X1, SSyIM) = Let C =
x in stages by a Henkin argument.
{ Co, Cl,...
be a new set of constants and let L
,...,cnIj.
arithmetic augmented by { go
S t a g e 0. S t a g e 2n
E
x
Let T0 in Lo be any completion in of PA. i1. Let TZni3 be TZn u { E % : x E Xn U{
Note that, if T2n
S t a g e 2n
be the language of
f
i2.
x,
then T2n+l
Let T2ni2
E
1 ;
f
n
x.
be any completion in LniI
%
:
x ?!
Xn
n
of TZnil
such that TZnil
x.
Let T = U Tn, M the term model of T (i.e. the model generated by all definable
n functions, using the constants of C).
Since M
b T 2 PA, M
is a model of arith
metic. The fact that
x ESSy@l) is clear:
Xn = {
X E 0:
TI
E
% 1
= { X E W:
; E % 1
Mk
n The converse follows also by construction: Let
b
E
IMI is explicitly definable from some c. MI= v u ( $ W E
c+
$*Gi ...c. v
for some $*.
E
SSy(M).
n @U0VI
,...,ci
"0
)
"k1 Thus, we can represent X f SSy(M) as
be any formula. Since any
kI
, we
have
1.
Lectures on Nonstandard Models of Arithmetic
for all m > 2i
Tm
E
x.
T~
[ z E w:
=
+ 2, i
Thus X
=
I
L
max{iO,
x.
E
...ci k  l 0 
xci
15
1
...,ikl 1 .
Thus, X is recursive in Tm, where
@ED
( Remark: The proof that i => ii is trickier. One does not have new codes to make elements of
x standard.
Thus, one alternates p a r t i a l z y completing the
theory first the C sentences, then I: sentences, etc. and representing 1 2 by using independent formulae of increasing complexity. ) elements of
x
Theorem 2.8 can be generalised. For example, one can replace PA by any consistent r.e. extension To (using Craig's observation that an r.e. theory has a recursive axiomatisation to handle stage 0).
A more important generalisation due indepen
dently to Don Jensen and Andrzej Ehrenfeucht (Jensen and Ekrenfeuckt 1 9 7 6 ) and David Guaspari (Guaspari 1 9 7 9 ) is the following: 2.10.
Theorem.
Let P be a consistent completion of PA in a language extending
that of arithmetic by a countable set of individual constants. Let able cclosed family of sets. i. Rep(!?) 5 ii.
x be a count
The following are equivalent:
X
There is a model M
b T with SSy(M) =
x.
The proof of Theorem 2.10 is a bit mare delicate. The extra constants offer no real problem they can be handled finitely many at a time. one only has R e p ( T )
c
x and not T x. E
The problem is that
However, for each finite n,m the Il
consequences of T referring only to the first rn constants is a set in
x.

Thus,
in the construction underlying the proof that i=> ii, one simply handles progressively larger chunks of T as well as progressively larger sets of constants.
I omit the details. Theorem 2.10 is, of course, of interest in its own right, but it is also a lemma. The theories T to which one wishes to apply the result are the complete diagrams of models.
As apparently first noticed by Guaspari, this allows an iteration of
Theorem 2.8 which, by a direct limit argument, yields: 2.11.
Theorem.
Let
x be a family of at most ,ql
sets of natural numbers.
The
following are equivalent: i. ii. 2.12.
x is cclosed X=
S S ~ ( M )for some M
Corollary (Assuming CH).
I=
PA.
x 5 P(w)
is cclosed iff
x is the standard sys
tem of a model of PA. 2.13.
Open Problem.
Can CH be eliminated as an assumption in Corollary 2.12?
The derivation of Theorem 2.11 from 2.10 is entirely routine and I omit it. is time to move on to another topic.
It
We will not leave standard systems for very
c. S M O R Y ~ K I
16
long:
they are of extreme usefulness.
EXTENSIONS OF MODELS (THE 1960s 11)
3.
Till now I have largely considered single nonstandard models. Another development of the 1960s was the study of the embeddability of one nonstandard model into another, i.e. of generalising the r8le of N o as the standard model to allow nonstandard models to be standard relative to other models.
There is a big
difference in the relation between a nonstandard model M and an extension N and the relation between N (NI

a
IN1 
E
0
and its nonstandard extensions: The new integers of
[MI could fail to be larger than all the integers of /MI. Thus we say
]MI is Minfinite if a > b for all b
E
IMI, and a is Mfinite otherwise.
There are three possible types of extensions of models (relative to the crude criterion of the existence of Mfinite and Minfinite integers): 3.1.
Definition. Let M _ C N be models of PA. i.N is an end extension of M, or M is an initial segment of N , E J N J  J M Jis Minfinite. N is a c o f i n u l extension of M, written M c N, if every a
written M c N, if every a ii.
E
IN1 
IMI is Mfinite. iii.
N is a mixed extensionofM if IN1  \MI contains both Mfinite and
Minfinite integers. All three possibilities are realised: 3.2.
Theorem.
Let M be a nonstandard model of PA. M has elementary mixed
extensions. 3.3.
Theorem. Let M
3.4.
Theorem.
b PA.
M has proper elementary end extensions.
Let M be a nonstandard model of PA. M has proper elementary
cofinal extensions. Theorem 3 . 2 is a triviality: Two new constants and compactness or any proper ultrapower provide the extension.
Theorem 3 . 3 is nontrivial.
It was first proven
by R. MacDowell and Ernst Specker in MacLbwelZ and Specker 1961.
I postpone a
proof of this Theorem until 57, when I will give a stronger result. section I will also give some related results for the countable case. is from Rubin 2962 and is relatively easy:
In the next Theorem 3 . 4
One doesn't have to construct the
cofinal extension directly; one merely extracts it from a mixed extension. The modern descendent of Rabin's extraction is the following.
3.5.
Theorem (Splitting Theorem).
Let
MzN
be models of PA. There is a unique
model k f f k PA such that M
Cc
?8* Ce N.
Moreover, the cofinal portion of the extension is elementary.
17
Lectures on Nonstandard Models of Arithmetic The Splitting Theorem is due to Haim Gaifman (Gaifian 1 9 7 1 ) and was the end product of a development begun by Rabin and partially furthered by A. Adler and Grigorii Chudnovskil.
It is a fundamental result in that it shews the crude
order theoretic trichotomy given by Definition 3.1 to be model theoretically significant:
Cofinal and end extensions are the, sotospeak, building blocks of
all extensions and it suffices to study these two extreme cases.
Moreover, it
points to a fundamental difference in the r6les of the two kinds of extensions:
3.6.
Corollary.
Cofinal extensions are elementary.
The reader can easily deduce 3.6 from 3.5. If M
elementary case of 3.5:
5
Another easy exercise is given by the
N, there is a unique model
4f such that M Sc1'8~
The full Theorem 3.5 is nontrivial it is equivalent to the formalisa
4 e N. bility within PA of the solution to Hilbert's 10th Problem.
Because of this, the
result is languagedependent; if one adds a few new predicates to the language, L the result can fail. If we let PA denote the extension of PA to a language L L L containing that of arithmetic assuming full induction in L and let A o , z I , etc. denote the quantifier classes of the extended language, the Splitting Theorem becomes 3.7.
L L Let M E N be models of PA and assume the extension is A elemen0 Then there is a unique fif such that M
If
M
Let $ n ~ ~ . . . v ~with  ~ , free variables as
E IMI.
 b $Fo...znl==> N b $ a o . ..anl
N iii.
To do this I need a lemma.
c$
M
is provably A,
k $zo..
$zo...an 1
in P A ,
iff N
b $Zo..
Parts ii and iii are immediate consequences of part i, which follows from the provable equivalence in PA of all C formulae with existential ones and the per1 sistence of existential formulae under extensions.
Proof
Of
3.5:
obvious:
Let M 5 N.
We want M
=
{ a
E
3b
=
a
E
INI: INI:
Moreover, it is easy to see that
Q
A function. 1
Ifif I
fif INI(a < b )
7
N.
The choice of
fif
is
is Mfinite }.
!ffis a
i.e. it contains 0 and is closed under segment of N,
E
structure for the arithmetical language,
I,+,*.
In fact, since
fif
is an initial
is closed under any PAprovably polynomially bounded total
c.S M O R Y ~ S K I
18
One A function that can be proven total and polynomially bounded in PA is GEdel's 7 @function: There is a A definable term @ for which the following are satisfied: 1
for each n there is a polynomial &(VO,...,unl)
i.
1 V U ~ . . . U ~ 3~V n 1 B(uo,vl) 5 V o
PA
ii. PA
c vvO...unl
iii. PA
&(u,,
$vo...u
3 V ~ . . . U ~  $~ u o...u
PA
iv.
Formally: We prove by induction on n that
since J, is quantifierfree. Since, moreover, a
E
Igfl was
$.
is trivial.
E
Observe 
v0 < b Wo,
by 3.8
arbitrary, we have:
(M"f,B) b v ~ o ~ o . Induction s t e p :
If n > I, there are again two cases:
Case 1.
$
E
3n. This is trivial.
Case 2.
$
E
Vn.
Using v above,
(M,B)
Write $
=
Vvo3
I= V U o 3 U l J , = >
ulJ, and let a
(M,f3)
I=
3VlVV 0
E
(4 f.B) 3VlJ,(Z,B(Vl,Z))
I=
and, since a
E
=> (4f,B) I= a u1 $Civ arbitrary, (4 ,@) / v U o 3 U l
lffl was
8'
$VoVl.
( N.B.
The essen
t i a l ~ n e w of the formula in question was based on contraction of quantifiers, which 2n
!ffdepends on the bound
i or a different ntupling function.
) &ED
The interested reader is referred to MZ&k 1982B for an axiomatic analysis of the proof of the Splitting Theorem (in the form 3.7) and to Motohashi A&B for additional observations on this result. I also refer to my survey Smoryn'ski 1 9 8 1 A
Lectures on Nonstandard Models of Arithmetic
19
for more on cofinal extensions; there is much of interest in them, but I do not have the space to devote to them here. I have given more space to the Splitting Theorem and given a more detailed proof of it than many expositors would because it is rather more important than it appears.
It does, as I said above, establish the significance of the crude order
theoretic trichotomy of extensions into cofinal, mixed, and end extensions, and it justifies focussing one's attention on the two extremes. Moreover, as first observed by Wilkie, it has its applications.
(Cf. Theorem 4 . 8 , below, or, for a
simpler application in the style of Wilkie, Smoryriski 1 9 8 1 A . )
I shall close this
section with an amusing application. The Tarskian attempt of the 1950s to turn model theory into algebra resulted in a number of preservation theorems theorems characterising those sentences preserved under various kinds of restrictions, extensions, and homomorphic images. With the emergence of the notion of an end extension came also the problem of preservation under end extensions. This was settled in Feferman and XreiseZ 1 9 6 6 by appeal to a manysorted interpolation theorem.
Straight model theory also
suffices to prove the characterisation, and, in the arithmetic case, the result is
an easy consequence of the Splitting Theorem: 3.9.
Theorem. Let L be a language extending that of arithmetic. Let
I@~...V
n 1
be an arithmetical formula. The following are equivalent: i. ii.
Proof:
for all models M c
Mb
&o...Fnl
PAL
c Qi
1
L
and a.
Nel= Go..., n1
=> for some ?formula

N of PA
,...,anl
IMI,
E
JI,
JI.
The proof that ii => i is routine.
i => ii. Suppose Qi is upward preserved under all end extensions. By Theorem L 3 . 7 , it follows that @ is preserved under all A elementary extensions. Letting 0 L L' extend L by a stock of primitives sufficient to make all A formulae equivalent 0
to quantifierfree ones, we see that @ is preserved under all extensions M 5 N of models of P A L ' . By a standard result of model theory, @ is existential in PAL' , L whence 1; in PA &ED
.
4.
SATURATION PROPERTIES (THE 1970s I)
The model theoretic notions of types and of saturated and special models crystallised in the 1960s.
Although a crude mixture of recursion theory and model theory
can be found already in the late 1930s with Bernays' arithmetisation of the completeness theorem and one can find further discussion of the complexity of models in the 1950s and 1960s, the refined mixture of the two branches of logic namely, recursive saturation only arrived in the 1970s.
As one might predict on the
c. S M O R Y ~ S K I
20
basis of the fact that recursive saturation is a marriage of model theory and arithmetic definability, this notion is of great importance in the study of nonstandard models of arithmetic.
In fact, recursive saturation of a sort was
being applied in nonstandard model theory before it was realised that such a concept existed.
In the present section, I will discuss these early applications
and one or two minor issues related t o recursive saturation; in the next section,
I will digress to discuss recursive saturation in a more general, but arithmetically relevant, context. First, of course, come some definitions: 4.1.
Definitions. Let M be a model for some (not necessarily arithmetic) lan
,...,
bmwl E 1Ml given parameters, and ~ U ~ . . . U ~  ~ U ~ . .nim1 .U a set of guage, bo formulae with only the free variables shewn. 
i. set T 5
M
7,

. ba~type  ~ over M if, for every finite The set ? ~ ~ . . . ~ ~  ~ b ~ . .is
k
(M
$v o...v
S U ~ . . . U ~  ~

b

n1 o..*bml
1
tJET The type 7 ~ ~ . . . v ~  j b ~ . . . b ~is  ~recursive if the set TU o...v
ii.
(or, rather, the set of its numerical codes rtJ1) iii.


The type ~ v ~ . . . ~ ~  ~ b ~ . . .isb ~a rtype, ~ for
(e.g. all formulae, Z formulae, TI formulae), if every
n
iv.
M b M k
B V o...v

The type ?v0.. .v nl 5o...bml
n 1
$
r
E 7
a set of formulae belongs to
r
is realised in M if
( ~ ~ o . . . v n ~ 1 5 0 . . . b ), i.e. for some
~o...Znlbo...Fml, for all
nim1
is recursive
m1
a.
,...,anl IM\ E
E 7
V. M is recursively rsaturated if every rtype over M is realised in M; M is recursivezy saturated if M is recursively rsaturated for the set r of all formulae of the language of M.
N.B.
In the arithmetic language, we can restrict our attention to 1types ‘ruz
with only one parameter because of the arithmetic ability to code finite sequences. We will be interested, however, in models of theories in languages not admitting such coding. One of the keys to the arithmetic importance of these notions is the following notion: 4.2.
Definition. Let
r
be a set of formulae. We say
r
a h i t s a truth definition
(more accurately: a satisfaction relation) in PA if there is a formula Tr (v such that, for all PA
where ( v 0 ,
$ V ~ . . . V ~ E~
r
r
b@U~...V~~
...,Unm1)
v ) 03
I
with free variables as shewn,
tf Tr (r$’,(vo,...,V r n1 ) ) , is the code of the sequence V o ,
...,V n1‘
The bearing of this notion on recursive saturation is the following lemma implicit in Robinson 1963 and very nearly explicit in Friedman 1 9 7 3 : 4.3.
Lemma (RobinsonFriedman Lemma).
Let M be a nonstandard model of PA and
21
Lectures on Nonstandard Models of Arithmetic suppose of
T,
M
r
admits a truth definition. Then: M is recursively rsaturated.
~ a recursive type over M. By the recursiveness Proof: Let b E IWI and T U be there is a formula Tau(u ) such that, for any formula @ u U o , @VV E T iff
0
0
Let, further, Trr be the truth definition for I.
Tau('$').
From the fact that
M I=
T
is a type, we have, for each
3 u 1 V r@'
ii.
1
obtained from M by an appropriate inter
lLo'
Let M be a countable recursively saturated model of T o .
Construct a recursively saturated model MI of Tl with SSy(Ml) = S S y ( M ) .
Letting
M
be the L reduct of M and appealing to Lemma 5.8 inside M we have SSy(M ) = 0 0 1 1 0 SSy(Ml) = S S y ( M ) . By Theorem 5.10, M is isomorphic to Mo. Pulling the structure of MI back to M yields the expansion desired. ii => i.
&ED
By Lemma 5.9.
Refinements: Obviously, To need not be assumed complete as T h ( M ) is both rich and complete. If T is not assumed complete, but some axiomatisation of T is coded 1 1 in S S y ( M ) , it has a completion so coded and the existence o f an expansion to this completion yields that for T I .
In particular, a countable recursively saturated
model M of a rich theory T is expandable to a recursively saturated model of
!2%(M) U T l , for any recursive theory T I consistent with T h ( M ) . This latter version of expandability to recursive theories has nothing to do
c. S M O R Y ~ K I
34
with richness; it is a basic expandability result called respzendence and due independently to J.P. Ressayre and to Jon Barwise and John Schlipf.
The present
sharp form for rich theories was also proven in Knight and Nude2 1 9 8 2 4 , where a generalisation to arbitrary theories in finite languages in terms of "ideals" is given. Resplendence the expandability of countable recursively saturated models t o recursive theories has many applications to models of arithmetic, particularly to recursively saturated models of arithmetic. Many of these applications, however, are not as powerful as those obtainable by more direct methods; I refer the reader to the papers cited at the end of the previous section for such applications. What I really want to do now is digress even further from the subject of nonstandard models of arithmetic and prove an analogue to Tennenbaum's Theorem. Recall that Tennenbaum's Theorem (actually, Corollary 1.6)
told us that no non
standard model of PA could have recursive operations of addition and multiplication.
By Theorem 5.1,
the
+
and reducts of a nonstandard model of ?A are
recursively saturated models of PSA and SMA, respectively; by Theorem 5.15, converse is countably true:
the
A countable recursively saturated model of ?SA
is the +reduct (reduct) of a nonstandard model of PA.
(SMA)
Hence, there are no
recursive, recursively saturated models of ?SA or SMA. What we shall see is that this is a better way of looking at Tennenbaum's Theorem it asserts the nonexistence of recursive, recursively saturated models of decent rich theories. Before proving the analogue of Tennenbaum's Theorem, we need two definitions. 5.16.
Definition.
Let I MI be M as identity. We say M
Let M be a model for a given recursive language L .
w and let equality, if it occurs in L , be interpreted in
is recursive7y presented if there is a recursive satisfaction definition for atomic formulae of L in M:
MI=
...,xn 1
The relation, for xo,
Go...Zn1 '
is a recursive relation of
r$Uo...V
' and
n 1
E
w and formulae I$,
(X~,...,Z~~) (the usual recursively
obtained ntuple). If L is finite, this amounts to each primitive relation and function being recursively interpreted in
W.
The notion of recursive presentation is the appropriate version of model theoretic complexity. 5.17.
The notion of decency of richness is the following:
Definition. A rich theory T is e x h t e n t i a Z Z y r i c h if the recursive sequence
of formulae witnessing the richness of 7' can be chosen to be purely existential. Examples will follow shortly. First, let me state the main result:
Lectures on Nonstandard Models of Arithmetic 5.18.
Let M be a recursively saturated
Theorem (Tennenbaum's Theorem Revisited).
Then: M is not recursively presented.
model of an existentially rich theory. Proof:
Observe first that S S y ( M ) , being completionclosed, contains a nonLet X be such a set and
recursive set e.g. some consistent completion of PA. let k M
E
35
( M I = u code X.
I= $nzo., .Kr where
Then: n
avo..
E
X
iff M
n
E
3 m0
iff
is the formula n
.V
n
k
E
,...,m p I
s.t.
v with$ quan'tzifier
n
free. Evidently, X is r.e. in any presentation of M . Since SSy(M) is closed under complementation, T X is also r.e. in any presentation, whence X is recursive in such.
Since X is nonrecursive, the representation is
&ED
not recursive. 5.19.
Examples.
The following theories are existentially rich and, hence, have
no recursively presented, recursively saturated models: i.
PSA; cf. 5.5.iii
ii. SMA; cf. 5.5.iv iii.
ODAG+; cf. 5.5.vi
iv.
RCF; cf. 5.5.vii.
RCF, at least two of +,.,
0.
it will remain relatively fixed.
p a r t i t i o n , o r y
€1
X,P.
We shall denote x by n in the following since
Lectures on Nonstandard Models of Arithmetic
Basis.
For n
31
1 , this is merely the assertion that a finite union of finite
=
In this case, Y is clearly recursive in X,P.
sets is finite.
Induction step.
Without l o s s of generality, we may assume X
=
w.
Let
(w;+,*,',O,P). By Theorem 6.1, SSy(N 0 ,P) is cclosed. 0 Choose (M,P), by Theorem 2.10, to be any nonstandard model of T h ( N ,P) with 0 SSy(M,P) = SSy(NO,P). The canonical embedding of ( N ,P) into (M,P) is clearly P:(nyl)
y and ( N ,P)
+
elementary.
=
0 Moreover, the restriction to w of any definable subset of or a rela
tion on (M,P) is definable in (No,P). Let a
E
Using a, deviously define the rela
\(M,P)I be any nonstandard integer.
tion R on wn+'
by
Rxo...xnlx:
P(xo,...,xn~l,x)
=
P(xo,...,xnl'a).
R is codable in (M,P), whence definable in (NO,P). Let @Vo...VnlV
Proof:
Suppose P ( X ~ , . . . , X ~  ~ ,=~ )i < y .
(M,P) I=
Go....: n I x


whence (No,P) b Go...Fnnlx
(No,P)I= v v ( Since p(x0
&o...FnIv
,...,
=
++
For any x E w, P(X0 Xn,,X) = P(xo Xnl,a) tf P(F0,. 'xnl ,x) = T , P(xo X ~  ~ ~ = X i.) Since x was arbitrary, P; o...xElv i whence (M,P) satisfies the same.
,...,
ft

define R.
.. ,...,
i, (M,P)I=
Proof of 6.3, continued:
1
,...,
,,
QED
I$LZ~...X~~U.
We enumerate a prehomogeneous set Z as follows:
... < znm1 be arbitrary natural numbers. n, such that for any subsequence Suppose we have generated z o < ... < zkIl k < z i < ... < zi , we have (No,P) b @zi .,.zi By the Claim, for each z zi . i0 0 n 0  n1 n ... < z i , we have (M,P) c. ...zi a. From this it follows by Theorem 2.1.i n 1 '0 n1 z for all that there are infinitely many z ' s in w such that (N ,P) $yi ...z 0 o in?
Let z o
ii). I f i f ) h o l d s t h e n f o r any P c o n t a i n i n g A say w i t h
= 4, e v e r y c o n j u g a t e o f p i s d e f i n a b l e o v e r M. conjugates.
Cow,
suppose iii)h o l d s .
number o f c o n j u g a t e s .
Then,
Thus
f o r each
T h i s i m p l i e s as i n Shelah1978
depends on a f i n i t e e a u i v a l e n c e r e l a t i o n o v e r A. t h e r e a r e lii
< k, e l e m e n t s o f
M such t h a t :
p
has a t most 4
b(F;y),
IAl
IT'
d 4 has a bounded
( c h a p t e r 111) t h a t d 4
Thus f o r any
M containing A
k b ( F ; E ) V<E(F,iii):i
< k>.
3. FREE EXTENSIONS O F TYPES OVER M O D E L S I N STAELE THEORIES.
Now,
a s t a b l e t h e o r y can b e d e f i n e d as one f o r which i f p i s o v e r A t h e n p i s
d e f i n a b l e o v e r A. by H a r n i k .
3.1
We u i l l use a p a r t i c u l a r y s t r o n g form o f t h i s r e s u l t e n u n c i a t e d
H e n c e f o r t h ue d e a l o n l y w i t h s t a b l e t h e o r i e s .
LEMMA ( C H a r n i k  H a r r i n g t o n
19821.
I f T i s stable,
f o r every formula
4
and each
p € S ( A ) t h e r e e x i s t L  f o r m u l a s e(!?;Z) and M 5 ; Z ) such t h a t :
i)
For some 3, e(X;3)
ii)For e v e r y 6 € A,
I=
€ p.
i f e(%;a) € p t h e n
[bt!?;6)
€ p iff
At6; B) 1.
Now we w i l l d e f i n e a f r e e e x t e n s i o n o f a t y p e o v e r a model and show t h a t t y p e s o v e r models a r e s t a t i o n a r y .
We a l s o p r o v i d e some o t h e r i m p o r t a n t
c h a r a c t e r i z a t i o n s o f f r e e e x t e n s i o n s of t y p e s o v e r models.
In the following
s e c t i o n we i n t r o d u c e an i m p o r t a n t concept and t h e n r e t u r n t o t h e t a s k o f d e f i n i n g
J.T. BALDWIN
78 f r e e extensions o f a r b i t r a r y types.
I f p i s a type over a model, M,
then we w i l l choose t h e f r e e extension o f p t o be
one which i s d e f i n a b l e over M .
Such an extension e x i s t s and i n f a c t i t has
several other n i c e p r o p e r t i e s which we record nou.
f o r some
ZI
ii)(THE FUNDAMENTAL ORDER) Then p
,
over A i f
b(ic;5) 8 p.
f A,
2 q,
and q i n S(B).
Let p be i n S(A)
i f every formula which i s represented i n p i s represented i n

5
q and q
5
n a t u r a l l y extend t h i s n o t i o n t o p
2
q.
p
i)The formula bCii;l) i s represented i n t h e type
3.2 DEFINITION.
We w r i t e p
q if p
p.
When C q
i s represented i n p i s represented i n o.
domp
fl
domq we can
i f every formula i n L ( C ) )
which
We denote t h e equivalence
c l a s s o f q by Cql.
3.3 D E F I N I T I O N . Let M be a model o f T and an h e i r o f p € S(M) i f q extends p and p coheir o f p on A i f t(A;M
3.4 THEOREM.
M be contained
,,,
q.
i n A.
Let a r e a l i z e p.
Then q i n S(A)
is
Then t(B;A)
is a
U 3 ) i s an h e i r o f t(3,M).
Let M be a model o f a s t a b l e theory T,
p f S(M)
and M
4 C,
then p has
a unique d i s t i n g u i s h e d extension t o a complete type over C which i s i t s  h e i r , coheir,
and d e f i n a b l e extension.
The proof o f t h i s can be found i n LascarPoizat o r i n any o f t h e various p r e p r i n t s o f Baldwin198x.
4.
STRONGLY SATURATED MODELS
We uant nou t o d e f i n e t h e f r e e extension o f a type p uhose domain may n o t be a
79
Strong Saturation and the Foundations of Stability Theory For t h i s our v a r i o u s n o t i o n s o f a l m o s t w i l l be u s e f u l .
model.
e x p l o r e one f u r t h e r concept.
Eut f i r s t ,
we must
T h i s n o t i o n was i n t r o d u c e d b y Shelah as
F a X  s a t u r a t i o n and p l a y s an i m p o r t a n t r o l e i n t h e c o u n t i n g o f models o f s t a b l e theories.
4.1
However,
The model M i s s t r o n g l y  X  s a t u r a t e d
DEFINITION.
X, any
0 then easily [O mod pnl z 6 P[F((t))l, so let y' = z in this case. If v(z) '< 0 find z 1 6 P[F((t))]
holds let z = y
v(zz 1 ) > v(z) and let
with
property Po(z).) treated,
OK
z' =
Replacing z
by
.
(If v(z) = 0, we use the [O mJd Pnl z', we may conclude by the first case
(22,)
by induction.
At this point we introduce the crosssection to get the effect of f u l l secondorder quantification over the value group 2 in a nonkaplansky field. For x C F((t)) let Z(x) be the set:
where
is the predicate corresponding to an additive polynomial P with
Po
P[F] # F. This Z ( x ) is an arbitrary subset of on N , the undecidability follows. S5.
N
.
Since we already have
+
A few open problems
Although we speak in term of decidability and undecidability, we are really just studying the structure of the definable subsets of F( (t)) or F(t). The hard questions concern definability in the pure language of fields, but there are also open problems in enriched languages. 5.1
The language of fields
1.
Is the constant subfield definable?
This for me is a key question. My investigations have produced nothing worth going into here. 2.
Is the predicate 'n is a power of p" definable?
This is a predicate on the value group. One can certainly conceive of this happening, although there is no concrete evidence for it. 2A. Is the theory of
F((t)) undecidable in the language of fields enriched by the above predicate?
This is extremely likely. Van den Dries points out that Presburger arithmetic enriched by the "pk"predicate more to work with. 3.
decidable, but we of course have considerably
Is the predicate "the leading coefficient of x is 1" definable?
This means x = iL&x)xiti field is finite, using
= 1.) This is trivially so if the base with x v (XI = 1 for a F F*. q
G.L. CHERLIN
94 3A.
If F is infinite (and perfect), is F((t)) undecidable in the language of fields extended by the above predicate?
This also seems extremely likely. 4.
Are there any nontrivial definable predicates on than those of the type catalogued in S4?
This, of course, is the real question. its accompanying comment. 5.2
F((t)), other
The reader may formulate question 4A. and
Ehriched languages
Let L(S), L(F) be the languages of valued fields extended by unary predicates S,F respectively, which will be interpreted as follows in power series fields F((t)) (for F perfect, infinite).
s = Itn: n t
x 1
F = constant subfield It turns out that in this context these languages are equivalent  more precisely L(S) has the same Odefinable relations as L(F,t)  and strong: they amount to the language of F enriched by quantification over countable subsets. The method used is a somewhat awkward elaboration of the coding in S2. no light on weaker languages.
It casts
5.3 Welcompleteness
Consider the language of fields extended by predicates interpreted in F( (t)) to mean:
where
R
(and x = (*)
R*(x,y,z,
...
)
varies over some extension of the language of fields xiti,...).
The following is conceivable:
If F is modelcomplete in the extended language then is modelcomplete.
This is true in any case in characteristic zero, and it does not contradict anything known in characteristic p. Of course such a statement would not be obtainable from a modeltheoretic trick. It is essentially a strong statement about algebraic geometry over F( (t)), in a rather vague form.
Undecidability of Rational Function Fields in Nonzero Characteristic
95
References Kochen, S., "Diophantine problems over local fields I, 11," Amer. J. Math. 187 (19651, 605630 and 631648. [2] Baur, W., 'Undecidability of the theory of abelian groups with a subgroup,". PAMS 55 (1976). 12518. [31 Becker, J., Denef, J., Lipschitz , L., "Further remarks on the elementary theory of formal power series rings," in Model Theory of Algebra and Arithmetic, Pacholski et. al. eds., LNM 834, SpringerVerlag, NY 1980, pp. 19. [4] Cohen, P., "Decision problems for real and padic fields," Comm. Pure Appl. Math. 22 (19691, 131153. [5] Delon, F., "Hensel fields in equal characteristic p > 0," in Model Theory of Algebra and Arithmetic, Pacholski et. al. eds., LNM 834, SpringerVerlag, NY 1980, pp. 108116. [6] Delon, F., "Quelques propri6tgs des corps valugs en thgorie des modGles," Thise d'Etat, Univ. Paris 7, 1982. 171 Duret, J.L., "Les corps faiblement alggbriquement clos non separablement clos out la propriit; d'indipendence,' in Model Theory of Algebra and Arithmetic, Pacholski et. al. eds., LNM 834, SpringerVerlag, NY 1980, pp. 136157. (81 Ershov, Yu., "On elementary theories of local fields," Alg. Log. 4 (1965), [l] Ax, J.,
530. [91 Ershov, Yu., "Cn the elementary theory of maximal normed fields, "Doklady 165 (1965), 13901393. [lo] Greenberg, M., "Rational points in henselian discrete valuation rings," Publ. I.H.E.S. 31 (1966), 5964. [ll] I. Kaplansky, "Maximal fields with valuation I, 11," Duke Math. J. 9 (1942). 303321 and 12 (1945), 243248. [12] Kochen, S . , "The model theory of local fields," in Logic Conference, Kiel 1974, LNM 499, SpringerVerlag, NY 1975, pp. 384425. 1131 Martyanov, V., "The theory of abelian groups with predicates that
distinguish subgroups and with endomorphism operations (Russian)", Alg. Log. 14 (19751, 536542. [141 P. Ribenboim, Th6orie des valuations, Presses Univ. Montreal, 1964. [15] Robinson, R., "The undecidability of pure trans. ext. of real fields" Zeitschr. f. math. Logik und Grundl. d. Math. 10 (1964), 275282. [16] Rumely, R., "Undecidability and definability for the theory of global fields," TAMS (1980), 195217. [17] Schilling, 0 . . The theory of valuations, AMS Math. Survey, 1980. [181 Serre, J.P., Corps LOcaux, Hermann, [19] M. Ziegler, "Die elementare Theorie der henselchen K&per," Thesis, K6ln 1972.
1. I have since found more precise information, described without proof in S5.2.
LOGIC COLLOQUIUM '82 G. Lolli, G. Long0 and A. Marcia (editors) 0 Elsevier Science Publishers B. V. (NorthHolland), 1,984
91
Remarks on T a r s k i ' s problem concerning
Q
, +,
,
*
exp )
Lou van den Dries
S choo 1 of Mathema tics The I n s t i t u t e f o r Advanced Study Princeton, New J e r s e y 08540 U.S.A.
.INTRODUCTION
,+ , * ) s t r u c t u r e @, +, * , exp)
In h i s monograph on t h e elementary theory of t h e s t r u c t u r e T a r s k i asked whether h i s r e s u l t s could be extended t o t h e ([T,
exp(x) = e x
p. 451). ( I n s t e a d of
,
b u t t h i s makes l i t t l e d i f f e r e n c e s i n c e
as t h e unique f u n c t i o n of t h e form
Tarski suggested the f u n c t i o n exp
is d e f i n a b l e i n
x b> f (ax)
t h e axioms mentioned i n [T, p. 57, n o t e 201 f o r from adequate, see e.g.
(R
(R
f (x) = 2'
,+ , * , f)
which is i t s own d e r i v a t i v e ;
Th@
,+ , , f)
are far
[DW].)
Before we d i s c u s s T a r s k i ' s question, l e t u s b r i e f l y review some a s p e c t s of
his work on
I, +,
(1) D e c i d a b i l i t y of ( 2 ) Tha,
+ , .) =
a)
Th@,
and see what use has been made of i t :
+ , ) ,
t h e o r y of r e a l closed f i e l d s
(3) Elimination of q u a n t i f i e r s f o r
@t
(4) P r o p e r t i e s of d e f i n a b l e s u b s e t s of (5) P r o p e r t i e s of d e f i n a b l e f u n c t i o n s
,
, < , 0 , 1 , + , .) , lRn ,
.
These a s p e c t s are c l o s e l y r e l a t e d i n T a r s k i ' s work, b u t i t makes sense t o d i s c u s s them s e p a r a t e l y .
(1) is a n i c e r e s u l t i n i t s own r i g h t and q u i t e u s e f u l
i n many t h e o r e t i c a l d e c i d a b i l i t y questions, b u t has otherwise n o t been important
i n s e t t l i n g open problems, a s f a r as I know.
(2) i s sometimes u s e f u l i n proving
p r o p e r t i e s of r e a l closed f i e l d s : i n c e r t a i n cases t h e only known proof c o n s i s t s of f i r s t e s t a b l i s h i n g t h e p r o p e r t y f o r t h e f i e l d of reals by transcendental methods and then invoking (2).
(This i s c a l l e d T a r s k i ' s p r i n c i p l e . )
(2) and (3)
combined g i v e a t r i v i a l and improved s o l u t i o n of H i l b e r t ' s 17th problem,
,
L. VAN DEN DRIES
98
and some important g e n e r a l i z a t i o n s , due t o A. Robinson. The c e n t r a l r e s u l t i n T a r s k i ' s work seems t o me (3) a s I hope t o i n d i c a t e in
t h e d i s c u s s i o n of (4) and (5) below.
(Also, (1) and (2) are easy consequences Concerning (4): t h e s i n g l e most
of T a r s k i ' s method of e s t a b l i s h i n g ( 3 ) . )
f r u i t f u l f a c t i s t h e s o  c a l l e d TarskiSeidenberg theorem: t h e image of a s e m i 
i s a semialgebraic
a l g e b r a i c s u b s e t of lRm under a semialgebraic map lRm >Elu s u b s e t of
.
Rn
C l e a r l y t h i s i s t h e same a s t h e e x i s t e n c e of a q u a n t i f i e r
elimination f o r the structure than (3). (R
Q
,< ,
+ , *)
( r ) r ER,
which is s l i g h t l y weaker
(Semialgebraic = q u a n t i f i e r f r e e d e f i n a b l e with parameters i n
,< , + ,
.).)Another important p r o p e r t y of semialgebraic sets i s t h a t they
have only f i n i t e l y many connected components, and t h a t each component i s a l s o semialgebraic'
;
see [ K ] f o r a n i c e use of t h i s r e s u l t .
The b a s i c f a c t about (5) i s t h a t a continuous semialgebraic f u n c t i o n Rn >R
t h i s follows
is bounded i n a b s o l u t e value by a polynomial function:
e a s i l y from ( 3 ) ; an important a p p l i c a t i o n occurs i n [ H 8 r , p. 2761.
A simple proof, due t o
r e s i s t giving one o t h e r b e a u t i f u l a p p l i c a t i o n :
K. McKenna, t h a t t h e inverse of a b i j e c t i v e polynomial map
a polynomial mal.
From complex a n a l y s i s we know t h a t
c l e a r l y the r e a l valued f u n c t i o n
z
+>
I cannot
p
1
p:En
>
En
is holomorphic, and
i s continuous and semi
Ip'(z)l
a l g e b r a i c , hence bounded by a ( r e a l ) polynomial f u n c t i o n ( i d e n t i f y i n g
E2n). Therefore, by L i o u v i l l e , How t o extend a l l t h i s t o
p Q
1
i s a polynomial map.
, + , , exp
is also
En
with
Q.E.D.
)?
I t seems t o me t h a t concentrating most a t t e n t i o n on t h e analogue of (l), t h a t i s , d e c i d a b i l i t y of the elementary theory, i s a waste of time:
consider
f o r example t h e perplexing problem of deciding t h e statements p(e, e
with
p EZ[X1,X2,X3,
...I
f r e e p a r t of the theory.)
.
e
e
, ee ,
...) = 0
,
(And t h i s i s j u s t a t i n y p a r t of the q u a n t i f i e r
A n a t u r a l 'exponential'
f i e l d ' does n o t seem l i k e l y (but see [vdD
11
analogue of ' r e a l closed
[DW]),so I d o n ' t expect
This p r o p e r t y i s n o t an obvious consequence of T a r s k i ' s work; see a l s o the end of t h i s Introduction.
Remarks o n Tarski's Problem Concerning (W,f, . , exp)
99
an a t t r a c t i v e analogue of (2) f o r our exponential s t r u c t u r e . More p l a u s i b l e problems a r i s e i n the attempt t o extend (3),
(4) and (5).
To e x p l a i n t h i s l e t us go back t o t h e r e s u l t t h a t each semialgebraic s e t has only f i n i t e l y connected components, each semialgebraic.
This follows from
2 C o l l i n s [C] i n which a new d e c i s i o n method for the r e a l s is constructed, much more time e f f i c i e n t than T a r s k i ' s .
us h e r e :
But t h i s e f f i c i e n c y aspect does n o t concern
we a r e i n t e r e s t e d i n C o l l i n s ' key geometric idea, which he c a l l s
" c y l i n d r i c a l decomposition"; i t i s p a r t l y an a l t e r n a t i v e to, p a r t l y a considerable sharpening of t h e
notion
of q u a n t i f i e r elimination.
what a c y l i n d r i c a l decomposition of a s e t
X cRn
I n (3.6) we s h a l l define
is.
For the moment, we
i s the d i s j o i n t union of f i n i t e l y many
only mention t h a t such a s e t
X
a c e l l being a s u b s e t of IRn
m bmeomorphic t o a space R
,
m
,< n
.
The following c o n s i d e r a t i o n s i n d i c a t e t h a t a l t e r n a t i v e s t o (naive) quant i f i e r e l i m i n a t i o n a r e q u i t e welcome i n the s i t u a t i o n we a r e facing.
FAILURE OF 'NAIVE ' QUANTIFIER ELIMINATION The example below shows t h a t the elementary theory of
QR
,< ,
+ , , exp
(r)r
) does n o t admit e l i m i n a t i o n of q u a n t i f i e r s .
In
f a c t , much more i s t r u e :
Let
Proposition. functions,
Fi:R
"i
>
(Fi)i R
.
I
be any family of
Then the s t r u c t u r e (R,
admits q u a n t i f i e r e l i m i n a t i o n i f and only i f each (Note: Semialgebraic = d e f i n a b l e i n (p
,< ,
Fi
( r ) r ER,
(total) r e a l
>R
by
i s the s u b s e t
of lR3
which i s obviously d e f i n a b l e
1.
The r e d u l t is a c t u a l l y due t o H. Whitney. See [& p. 1101 f o r an e l e g a n t proof.
L. VAN DEN DRIES
100 Claim. 
There a r e no ( r e a l ) a n a l y t i c functions
an open b a l l i n IR3
,
0
centered a t
such t h a t
boolean a l g e b r a generated by the s u b s e t s Proof. and
The c r u c i a l f a c t s about
i s n o t an "algebraic" function.
f
i = 1,
...,k ,
must vanish on
+0
Fi
nU
f o r each
d.3)
(Fi = 01 of
U
,
i
G(f)
and
nu
,
,
Fi:U >IR
for all
Pd(x,y,f(x,y))
= 0
A t l e a s t one of the c E G(f)
Fi
nU
,
,
with
would c o n t a i n a whole neighborhood of
..., Pd a homogeneous polynomial of n U . Then f o r a l l 0 < X < 1 we have
+
d
.
XPl(x,y,z)
Take
d
+
with
2 XP2(x,y,z)
Pd
,
(x,y) E R x l ? '
for a l l
# 0
+
,
and
... ,
hence
and we see t h a t f
would be an a l g e b r a i c
o
We model the proof of the p r o p o s i t i o n on the argument j u s t given. f a n x I?>
IR be a n a l y t i c and d e f i n e
f (XI,.
..,xrrcl)
G(f)
i s quantifier free definable i n
= xrrclF(xl/xrrtl,.
.
. , X ~ / X ~ + ~ )
@
.
Let
1R by
If we assume t h a t i t s graph
, < , (r)r ER, + , * ,
P(x,f(x)) = 0
variables with
s u i t a b l e value
X >0
analytic function
for the
FXSZn >IR
n+lst
,
s a t i s f i e s i d e n t i c a l l y an equation polynomial i n
n+l
variables.
a l g e b r a i c function, hence
.
3 , We assume here t h a t converges on U
U
F
for a l l
x EXn X R ?
0
.
,
(Fi)i EI)
then w e d e r i v e e x a c t l y as before t h a t t h e r e is a nonzero real polynomial n+2
F1
say
and
= Po(x,y,z)
Pd(x,y,z) = 0
>
,
+ P1 + P2 +
(x,y,z) E G(f)
Let
0 = F1(Xx,Xy,Xz)
F SZn
.
f(Xx,Xy) = M(x,y)(X>O,y>O)
(otherwise t h e r e would be
F1 = P 0
(Xx,Xy,XZ) E G(f)
function.
U
belongs t o t h e
Suppose t h e f u n c t i o n s
i s i d e n t i c a l l y zero.
G(f)
3 c E R ). Write degree
are:
nU
,
have the property we want t o r e f u t e ; we may of course assume
t h a t none of t h e
Fi(c)
f
G(f)
> 01 ,
{Fi
...,Fk :U >a
Fl,
P
in
Substituting a
v a r i a b l e , we derive from t h i s t h a t the
given by (x,
,...,xn> &>
Q(x F (x)) = 0
' A
,
x Enn
xF(xl/X
,
The l e m below shows t h a t then
Q
,...,
xn / A )
a nonzero r e a l FX i s a semi
i s a l s o semialgebraic.
i s taken so s m a l l t h a t t h i s Taylor s e r i e s of
F1
Remarks on Tarski's Problem Concerning (lR,+, . , exp)
kuima.
If a continuous function g d >IR
equation Q(x,g(x))
'= 0
,x
101
satisfies identically an
E Rn , where Q is a nonzero real polynomial in
n+l variables, then g i s semialgebraic. From the results on pp. 106110 of [El, it follows that Rn can
Proof.
be partitioned into semialgebraic subsets Ao,A1, connected such that if x E .A 1
< i 5 m
and x E Ai
functions pil(x) < Pij:Ai >lR
,
,
then Q(x,Y)
...,Am
of x
... Am
vanishes identically, while if
the real roots of Q(x,Y)
... < Pik(i)(x)
with A1,
.
are given by continuous
(Obviously, these functions
are semialgebraic.) By continuity and connectedness, g must
coincide on each Ai (i
> 0) with one of the functions Pij
,
hence g
... U Am is semialgebraic. Since every p .A is the limit of points in A1 u ... U Am , the value of g at p is semialgebraically determined by its values on A1 u ... u Am . It follows that g is semirestricted to A1
algebraic.
u
0
The proposition forces us to look for new ways of solving (the realistic part of) Tarski's problem. A line of attack which seems quite promising to Define a kmanifold, k EN
En , n 2 k
,
to be a real analytic submanifold M of some k equipped with an analytic isomorphism h : M "  I R We introduce
.
an IRalgebra
classes ~
goes roughly as follows:
,
of kmanifolds, and for each manifold
for each k a class Jf(k)
M E &@k)
me
(
strutted in stages.
~
&(M)
of
(real) analytic functions on M'.
for1 each M (N.B.,
introduce a l l of M ( M )
,
The
the algebra d ( M ,) are con
as soon as we have M
,
we do not necessarily
at once.)
At stage 0 we introduce all semialgebraic kmanifolds and all semialgebraic (analytic) functions on them.
To obtain new manifolds, and functions
on them, as well as new functions on the manifolds already available, we use several constructions of which the following three are the most important.
Let
M
9 kmanifold already available.
102
L. VAN DEN DRIES
rf
(i)
are a l r e a d y a v a i l a b l e , then we introduce &he
f,g:M >R
kmani f o 1d graph(f)
t h e k+lmanifold
def (m,f)M ,
(f,g)M
the k+lmanifolds If
(ii) f
0
fo,
...,f d :M >R
+ fl Y +...+ ,
graph(f)
d fdY
(f,g)M
,
def
(x,f(x)) : x E MI
{(x,y) : f ( x ) (f,m)M
,
R
Assume f i r s t t h a t there i s a s u b i n t e r v a l
is f i n i t e .
hence t h e J
i s an i n t e r v a l and
i s continuous i n a t l e a s t one p o i n t of Proof, 
J
I cIR
If
with the property
b
such t h a t
n f'(b) J'
,
J
n f'(b)
is infinite,
must c o n t a i n a s u b i n t e r v a l
hence
f
J'
of
i s continuous a t each p o i n t of
I n the remainder of t h i s proof we s h a l l assume t h a t each s u b i n t e r v a l of has i n f i n i t e image under of closed segments f[an,bn]
f
.
[an,bn] c I
with
0
< bnan
g
k+l ( f , g ) X i s homeomorphic t o R
2
be
.
c o n s i s t s of a l l
are definable i n
define
C(X)
.
f
c o n s i s t s of a l l graphs
(or
k X i s homeomorphic t o R
there is o n l y t h e empty sequence
n = 0:
we l e t
and we p u t
W
corresponding c o n s t a n t f u n c t i o n s defined on
f
(1.6).
and functions between s u b s e t s of
33"
t o d e f i n a b i l i t y by an L  f o m l a i n the s t r u c t u r e
u
We f i x once and f o r
which i s of f i n i t e type, c.f.
@,
,
F(nl)
E
X
{XI
,
since ;
F(O) =
we let
1x3
ho),
and the only
be t h e only member of
.
Dec(X)
X
E
i n t o s e t s a l s o belonging t o
( f i n i t e ) p a r t i t i o n of
of
in
a decomposition of
For
Let
(%'
hx:X Z h(X) : ( y , r )
morphism
f
then
FOCI
Y be t h e p r o j e c t i o n map
Y
h(X)
,
dim X = k
belonging t o
h :Y Z h(Y) has a l r e a d y been defined. let
.
F(n'l)
then one can r e p e a t t h e same c o n s t r u c t i o n with
homeomorphic t o an open s u b s e t of JRk
dim(X)
E
h(X)
and suppose and
f E C(X)
l i f t s t o one of
>
X
G(f)
Dec(X)
. ,
has been defined f o r a l l
Then each f i n i t e p a r t i t i o n 1 namely (TT'(A1), ...,Tr (Ad)]
i s the p r o j e c t i o n on t h e f i r s t
n
coordinates.
X
E
F(n)
g ={A1, ,
. ...,Ad]
where
Let us c a l l t h i s
Remarks o n Tarski's Problem Concerning (R, +, . , exp)
'lifted' partition
gf.
Then we define Dec(G(f)) for f,g E ?(X)
The definition of Dec((f,g)X) complicated. Let again
9 E DecO() .
Then
=
B={Al,..., Ad
111
(gf 1 9E Dec(X)3
,
f
4 g
,
.
is a bit more
be a decomposition of X
,
i.e.,
9 induces a partition
E(f(Al,g(A
,...,(flAd,gIA )Ad 1
A1
of
(f,g)x
,
which we denote by
A second kind of partition of
,...,fm E c"(x)
fo,fl
partition of
with
(f,g)X
and the sets G(fi)
gfTg . (f,g)X
f = fo < fl
a
is d e f i n a b l e , so by the i n d u c t i o n hypothesis t h e r e i s a f i n i t e p a r t i t i o n of i n t o d e f i n a b l e s e t s on each o f which partition via o f which
f
w e g e t a f i n i t e p a r t i t i o n of
h
X
i n t o d e f i n a b l e s e t s on each
The set
Y = {x
dim X = N1
E Xlf is
.
use (b) f o r
n = N1
Y
.
X\Y
continuous a t
x)
contained i n
Y
,
so
which a r e contained i n
i s dense i n
Y
t o g e t a decomposition g o f lRN'
The s e t s i n
9which
a r e contained i n
X
arguments used i n t h e c a s e
dim X
< N1
and a r e open must be
< N1
The sets i n
,
9
and so the
apply.
To prove t h e claim we t a k e any nonempty open s e t i s continuous i n a t l e a s t one p o i n t of
.
which p a r t i t i o n s both X
but n o t open a r e of dimension
X
i s definable, we can
i s continuous o n each of those sets.
f
.
Then X i s open i n
I f we accept t h i s claim f o r a m m n t then, s i n c e
and
I f we l i f t t h i s
is continuous.
W e a r e l e f t with the case Claim. 
is continuous.
fh'
X'
U
U C X
.
To show t h a t
w e use e x a c t l y t h e same arguments
f
Remarks on Tarski's Problem Concerning (lR,+, . , exp) The r o l e of the decreasing segments
a s i n the proof of (2.1).
113
i s now
[an,bn]
U
of course taken over by a decreasing sequence of closed b a l l s contained i n
their diameter tending t o
11 (xl ,...,%1) 11
by the norm b a l l s are
@ere we l e t the distance on rPN' = max (lxll
,...,IX+~~) ,
be defined
i n order t h a t the
Rdefinable.)
m.
(3.9)
.
0
,
Assume t h a t
N
Then any two decompositions
>1
n = N1
and statement (b) of ( 3 . 7 ) holds f o r
gland g2of
EN have a COrrrmOn refinement,
t h a t is, there is a decomposition of RN p a r t i t i o n i n g each of the sets i n
2B1 u s 2 .
Proof.
Note t h a t RN =
(=,& N
s t r u c t u r e of decompositions of R t h a t @) holds f o r a conmmn base
where
n
= N1
see (3.5).
.
.,Bd]
qi
n1
is a decomposition of B
Bij
,
fh
,
and
,
g* p a r t i t i o n s
in
(=,")
,
,
are definable and continuous.
we have:
Bi Bi
= ("Jfl/...lfpl)Bi
, gCr:,Bi  3 3 R
Fix a set
(proper) decompositions of
By the assumption t h a t (b) holds f o r that
gland g2have
.
can be partitioned i n t o f i n i t e l y many definable sets any
This gives us the
So we can use the assumption
to reduce t o the case t h a t
9 and say gli,32ia r e
fh
,
.
X =lRwl
9 ,say
g=[B1,.
where the
where
B
Now c l e a r l y
Bi
such that, given
is
< $(x)
either for a l l
x
E
Bij:
fh(x)
o r for a l l
x
E
Bij:
fX(x) = g,(x)
or for a l l
x
E
Bij:
fh(x)
n = N1
each of the sets
Bij
there i s
> g,,(x)
g*E DecCpN')
. Now it should
,
,
.
such
b e c l e a r how to
.
L.VAN DEN DRIES
114
the
gp )
f X and
g* (and using r e s t r i c t i o n s s e t i n glU g2. [I
RN w i t h base
c o n s t r u c t a decomposition of
which p a r t i t i o n s each
of
Note t h a t Lemma (3.9) i m d i a t e l y extends t o a f i n i t e c o l l e c t i o n of de
.
N
compositions of lR
It i s t h i s s t r o n g e r form t h a t we s h a l l use i n t h e
f i n a l s t e p which follows now.
e. Suppose
(3.10)
statement (b) holds f o r Let
Proof. consider
N1
Ai
Formula f o r
>
B c n(Ai)
f(B,l)
gB % (I
n = N
Then statement (b) holds f o r
...,Am
A1,
of IRN
be given.
nRN >lRN'
where
n = N
and
.
Fix an Ai
and
i s t h e p r o j e c t i o n on
We a r e going t o apply (1.9) t o t h e d e f i n i n g We a l s o use
These c o n s i d e r a t i o n s give us a decomposition
partitioning
there is
n Ai
.
t a k i n g i n t o account Remark (1) following (1.9).
,...,f(B,k):B
nl(B)
statement (a) of (3.7) holds f o r
rr(Ai)
t h e hypothesis of t h e lemma. . o f lRN'
,
d e f i n a b l e sets
coordinates.
,
1
n = N1
a s l y i n g over
Ai
the f i r s t
N
n(Ai)
and such t h a t f o r each s e t
B E
si
with
k = k(B) and t h e r e a r e d e f i n a b l e continuous f u n c t i o n s with
>lR
f(B,1)
@ E Dec@RN)
of
u
This completes t h e proof of Theorem (3.7). (3.11) Corollary.
Each d e f i n a b l e s u b s e t of
Rn has only f i n i t e l y many connected
components, and each component i s a l s o d e f i n a b l e .
Remarks on Tarski's Problem Concerning (IR,+, . , exp)
Proof.
By (3.7) each definable subset
decomposition
9of
.
lRn
A
of
Rn
is partitioned by a
Now each of the f i n i t e l y many s e t s i n
(definable) c e l l , hence connected.
Each component of
of f i n i t e l y many c e l l s belonging t o
9,
115
9i s
a
i s therefore a union
A
and i s therefore definable.
( 3 . 1 2 ) Remarks (1) Theorem ( 3 . 7 ) was derived under the assumption t h a t
9i s
arguments i n ( 3 . 2 ) expansion
2 of
of f i n i t e type.

f i n i t e type
T h ( 2 ) ). Conversely, a weak form o f ( 3 . 7 )
(which i s r e a l l y an assumption on implies t h a t
9i s of
To be precise:
a l l of the definitions and
( 3 . 6 ) make sense and go through without change f o r any
@,a
is
(defined
Remarks on Tarski's Problem Concerning (lR,+, . , exp)
Proof.
W e s h a l l j u s t treat the case of
f'(a+)
.
117
(The case of
f'(a)
i s handled s i m i l a r l y . ) 4 = l i m i n f f(a+h)f(a) hL0 h
Suppose
r
Choose a r a t i o n a l h
>0
that
such t h a t
t h e s e t of a l l
h
contradiction.
0
w.
(A.2)
2
rh
.
>
0
such t h a t
Suppose
i s an i n t e r v a l ,
I
increasing, and i t s inverse
Proof.
the maps
Suppose f ' (a')
continuous on a l l of
Proof.
with
all
I
on (A.4)
J
,
(Note:
a EI
.
Then
f
f(1)
l/f'(a+)
is strictly
has the
if
f(a) = b
.
the c o n t i n u i t y assumption cannot
,
a
.
Then
+>
f '(a)
E P and an i n t e r v a l
J
I
hence
x E J
g
If
I
for a certain
around
a
a
E
I
a E I
.
g'(x+)
. .
Then
such t h a t
Then t h e continuous function
has t h e property t h a t
>0 ,
g'(x)
g : J 3P
< 0 for
would be both s t r i c t l y increasing and s t r i c t l y decreasing
by the previous lemma.
m.
.
for all
f ' ( a + ) = f'(a)
f'(a)
for all
is continuous, and
i s continuously d i f f e r e n t i a b l e on I
f
>
f'(x)
f : I >R
a r e w e l l defined, r e a l valued, and
f'(a+)
g(x) = f(x)cx
,
i s continuous and
defined on the i n t e r v a l
It s u f f i c e s t o show t h a t
>c>
x
fdl
f : I >lR
for a l l
I i s an i n t e r v a l ,
Suppose the contrary, say c
such
0
a I>
there is
>0
< r h is IRdefinable,
i s defined and equal t o
f")'(b+)
Lemma.
f'(x+)
>0
L e f t t o the reader.
be omitted.) (A.3)
b u t a l s o a r b i t r a r i l y small h
f(a+h)f(a)
i s defined ( i n IR ), and
property t h a t
Then t h e r e are a r b i t r a r i l y s m a l l
This s i t u a t i o n is incompatible with the f a c t t h a t
A
f'(a+)
,
< rh
f (a+h)f(a)
f(a+h)f(a)
.
a< r f'(x)
and
is
(This uses again ( 2 . 2 ) . )
I n c a s e (1) the i n v e r s e for a l l
E
x
f'
of
f
satisfies
(f')'(b+)
is . constant, c o n t r a d i c t i n g i t s i n j e c t i v e n e s s .
f'
case (2) we can apply t h e same argument a s i n t h e proof of Lemma ( A . 3 )
Then a.
f'(a+)
and
a
b> f'(a)
i s continuous, and by
f a
E
I
.
But then t h e
a r e Rdefinable,
hence
0
piecewise continuous, and t h e r e s u l t follows i m e d i a t e l y from ( A . 3 ) .
(A.6)
to get
i s piecewise continuously d i f f e r e n t i a b l e , t h a t i s , t h e r e a r e
on each s u b i n t e r v a l
(A.4)
In
0
a contradiction.
(A.5)
=0
= (f')'(b)
Corollary.
i s piecewise
t o i n c r e a s e with
Proof.
Suppose Cn
,
f : ( a , b ) >R
f o r each
n
6N
.
i s IRdefinable,
a
(The number of 'pieces
1
be a c l a s s of
dim(X)
.
C?
= Autlf(a)l(M)
=AutA,nB,(M).
111. C'est Shelah, dans 141, qui a introduit le premier des Ql6ments pour repr6senter les classes d'6quivalence modulo une relation dgfinissable, et il leur a donne le nom d'Ql6ments imaginaires. Ainsi il semble que si une thQorie satisfait la condition (l), ces Q16ments imaginaires ne soient pas nQcessaires : ils sont d6j2 en quelque sorte dans le modhle. Poizat dans C3l gtudie une condition de ce genre, qu'il appelle"6limination des imaginaires". Ce que nous aimerions montrer, s o u s l'hypothese ( I ) ,
c'est que :
Pour tout n E w , R(x,;) il
relation d'gquivalence sur
#,
ddfinissable, et
a
E
Mn,
existe un ensemble B c M , fini tel que
(autrement dit, B
et la classe de
a modulo R
sont dgfinissables l'un sur l'autre
Mais ceci n'est pas tout2fait vrai : prendre pour T la thQorie d'un ensemble
D. LASCAR
126 infini et
En fait Poizat s'est heurt6 a la mdme difficult6 et a d 2 introduire une autre condition : "l'glimination faible des imaginaires". Pour nous, c'est 11 que la condition (2) interviendra :
En fait, il est facile de voir, par compacit6, que n et k ne d6pendent que de
,Wk)
R, et que l'on peut aussi supposer qu'il existe une formule $(;
b.
sont exactement les suites satisfaisant
telle que les
$(;,b).


Esquisse de d6monstration. L'id6e est de trouver deux suites a l et a2 telles que {f cAut(M)
; M f=R(a,f(a)))
=
al
et M !=R(:,;~)
a2
AR(~,;~).
Mais ceci n'est pas toujours possible : prendre la th6orie d'une fonction 2 2 ff x [ s ( x ) # X A s (x) # X A s (x) =XI et pour
unaire s avec comme axiome
2
R(x,y) = C x = y v x = s ( y ) v x = s (y)].
Mais c'est presque vrai :
Pour A c M , notons Aut f (M) (le groupe des automorphismes Aforts) le sous A groupe engendr6par u {AutN(M) ; AcNxM}. I1 faut penser 1 ce groupe comme 6tant l'ensemble des automorphismes laissant la cleture alggbrique de A fixe. C'est pr6cis6ment le cas si T est stable et si l'on a rajoutg les 616ments imaginaires. Dans le cas gdngral, il faut consid6rer la cl6ture alg6brique dans un sens plus Btendu. Si a est un 6lgment imaginaire (i.e. la classe
a € Mn modulo une relation
d'gquivalence dgfinissable sur Mn), on peut aussi d6finir Aut (M) et Aut f (M). De mdme on peut dire que 1'616ment f3 (r6el ou imaginaire) est dgfinissable sur a si Auta(M) cAut (M), et que B est algdbrique sur a si
B
n'a qu'un nombre fini de
conjugu6s par les 616ments de Aut (M). Cela est dquivalent B dire que Aut f,(M)
c
Autg(M).
Maintenant, les r6sultats de C23, (th6orSme 614) nous permettent d'affirmer
127
Sous Groupes d'Automorphismes d'une Structure Saturk qu'il existe
al
et
a2 dans Mn
t e l l e s que M k = R ( a , a l )
et
nR(a,a2)
< A u t f (M) u Aut f (M) > = Aut fa(M) a2
al s i a e s t l a c l a s s e de
Soient
a modulo
R.
a l o r s A l e t A2 l e s c l B t u r e s a l g e b r i q u e s de
= < A u t f (M) u Aut f  (a) > al a2
Aut f,(M)
I1 s u i t donc que
c1
c
est dzfinissable sur A n A 2 , 1
CURULLAIRE 2. A w e c
b&
...,bk
l e s conjugu6s de
Le.4 rnCme.4 hypofh2hen que
a2.
On a a l o r s
/ h v h G l = h V ( k G 1 + $ 1) VFl+Kl
=
1
9. Meaning, Truth and D.A.S.
It is possible a non Tarskian analysis of theconcept of truth. Let us take as an example Peano Arithmetic. It is clear that there is point in introducing a concept of truth for its propositions only if we consider it applied, which happens if we use it, for example, as a metatheory of formal systems. Wishing now to introduce a concept of truth, it seems plausible to introduce it only for formulas that we wish to consider "meaningful". Which subset Po of the set of arithmetical formulas, do we wish to consider to be the set of meaningful propositions? We can propose to include in such a set exactly those propositions recognised by the theory itself, as it were, as verifiable or falsifiable, i.e. the p for which one of the formulas: k P jr6) ITP 3 +6& is verified, where T is the ordinary predicate "Theor". I shall not provide all the details; developping the idea and using a simplification due to A. Ursini one arrives at proposing as a set Vo of the "true and
Algebraic Logic and Diagonal Phenomena
143
meaningful" propositions the deductive closure of the set: L p : not I q p and I i p i ? ( r ~ 3 ) This deductive turns out to be in the arithmetical hierarchy. With usual techniques it is not difficult to "express" Vo with a suitable predicate bo and the following conditions p i Vo; Vo( $' ) c Vo turn out to be equivalent, thus, this concept of truth is not subject to Tarski's limitation Of course there are still some limitation of Godel's type:
z2
a) by Feferman's lemma there is a p with: (analogous of Godel's first theorem) Ip H 7 Go ( 3) ..
b) 7 i O ( . O #
Vo
(analogous of Gddel's second theorem)
. 
# 0 ' ) are meaningless, so in a way every But both the p in a) and ,VO('O discrepancy cases which leads us to distinguish between theory and metatheory. From an abstract point of view, forgotting all possible "phylosophical" meaning, it is interesting the construction of Vo from T, ? and this construction can be iterated obtaining for P.A.:
(Aldo Ursini 1976). The results obtained from myself (R.Magari 1975) and A.Ursini can be extended to every theory which "expresses" a predicate "Theor" and the natural ambient for this study are the D.A. Let A be a boolean algebra, S its dual space (think A as the algebra of the clopen sets of S), e a diagonal operator on A, Z = Y C Y , < the inverse of the dual relation of ;J Of course the analogous of T is now i.1; and the anal2 gous of i. is E Now theanalogous of V is the boolean filters, F generated by: M = i p i A : p # O a n z fp rvp;=ipkA:p#Oand 6 p g p i . as in the following example: It is possible that F be improper (
.
.
,T;p,q)(Tq we have Pq f 0 . 9.3 A is sem. cons. iff 4 is upper filtered. Now for investigate on F we can take theclosed set of S , C = n F = A M = = n : p : P + ~ , p ) Tp5. It is useful the lemma 9.4 < S, 4 > is inductive. Now we have an useful classification. 9.5 (i) There are in < S , d > at least two terminal distinct points not associated in 4 In this case C = 0. This is the only case of semantic inconsistency. 9.5 (ii) There is only one terminal point, a, and a {a: in this case C=ia] d A. 9.5 (iii) There is only one terminal point a and (a a: in this case C= {a$ f A. 9.5 (iv) The number of terminal points is > 1 but they are a l l associated in 4 in this case C is the set of terminal points and no subset of C is in A.
.
R.MAGARI
144
Now, M sing the techniques of 56 we can "formalise" F putting: qi. is.) = : o + 9.6l+p=C.p+
v
a=Q
...where
This is possible, of course, if the convenient families have suprema and ( .;(p.m qi ) + infima. Of course if j = ) i r we have jp = 7 p + ( .,' 7 qi + 'qi))) = l... The search for validity of: 9.7 If p~ F then ip c F and the search for compatibility of F with the congruence ry associated with F goes at further distinction in the case (ii) of 9.5:
xi
9.5 ii.1 There exists in S a b < a such taht, for every point x < S  (a,bj , x < b . 9.5.ii.2 Otherwise. We can proof that 9.7 is valid for all except 9.5 (ii,l) with b p. For compatibility with we have: if (i) (semantic inconsistency): Yes, but T=@,k is the id. 11 (ii.1) N O : a c P + a, a FO " (ii.2) Yes, but F is trivial: PO = fi1 = 1 " (iii or iv) (semantic consistency and Luconsistency: now we will call these algebras "regular") : Yes.

4
+
10. Progressions of Boolean Algebras with hemimorphisms.
7's
For hemimorphisms 7 with ~T we can generalise the previous results. We can study also thefinal product of iteration of the procedure and we find the following possibilities: (i) The final algebra A, is trivial (0=1): In this case 'c = 3 el and A, is in the case 9.5(i) (ii) A has two elements. is the total relation; Form a logical point of view this means (iii)C=S, that C 1=1 and for p f l c p = 0.
LOGIC COLLOQLIIUM '82 G. Lolli, G. Long0 and A . Marcia (editors) 0 Elsevier Science Publishers B. V. (NorthHolland), 1984
145
ON LOGICAL SENTENCES IN PA
Saharon Shelah Department of Mathematics The Hebrew University, Jerusalem, Israel Department of Mathematics Ohio S t a t e University, Columbus, Ohio, USA I n s t i t u t e ofAdvanced Studies The Hebrew University, Jerusalem, Israel Department of Mathematics University of California, Berkeley, Calif
.,
USA
Contents 5
1 . A representation of PH
5
2. On alcomprehension axiom 1 [We suggest a solution t o ? / n i  C A o
g 3 . A true 5
IT:
= ParisHarrington/PAl
sentence in PA, n o t provable in PA.
4. On theories with incomparable consistency strength. [We show how t o produce such reasonable theories. We also draw the reader's attention t o reasonable examples where theconsistency strength areequal b u t t h e r e i s no interpretation].
ii 1 . A representation o f PH
We give in t h i s section a representation of Paris Harrington [PHI r e s u l t s , in a way which will be helpful 1.1. Definition: An
F
E
1)
M9"
2)
M
L,
later.
(L,n)model i s a sequence
i s an Lmodel
I' Me
9"
5
n > such t h a t :
except t h a t functions a r e p a r t i a l (so M9" # 0 ) .
i s a submodel of M,+1
F
M = <M : 9"
M
e+ 1 ( f o r a . + l
5
n)
b u t f o r every function symbol
i s a t o t a l function (with range 5
I am very grateful t o Leon Henkin f o r saving the manuscript and t o Annalisa MarcJa f o r taking care of i t s typing. The author would l i k e t o t h a n k the NSF and the United StatesIsrael Binational Science Foundation f o r p a r t i a l l y supporting t h i s research.
S. SHELAH
146 1.1 .A. N o t a t i o n : Let
dp(cp)
M =<Ma:l
a
5
5

n>,
1
v and 1 $ 1
be t h e q u a n t i f i e r depth o f
@(X)
1.2. D e f i n i t i o n : F o r a f o r m u l a
E
L
only,and o n l y atomic terms used,w.l.o.g.) and an (L,n)model
(D
o f l e n g t h > dp(v)
5
Mri~.il = <M,:i.
n>.
i s t h e l e n g t h o f J,
.
( w i t h n e g a t i o n i n f r o n t o f atomic formulas
a
and a sequence (i.e.
n
t
dp(cp))
a(;)
M,,
from
= t(i),
we d e f i n e when
by i n d u c t i o n on t h e q u a n t i f i e r d e p t h o f cp :
fiI=cp[a] if
fi
: a.
M. = <Mi,,
atomic
: as u s u a l ( n o t e t h a t o n l y a t o m i c terms were used, and
t h e l e n g t h of
fi
i s > 0 = dep(cp),
hence we can compute
t h e terms and check t h e s a t i s f a c t i o n o f t h e r e l a t i o n in
1.3.
Claim: 1 )
a)
h a s a dp(+)model
IJJ
I f a sentence
b) u, has an nmodel
2)
J,
3) I f
fi
M,)
$
has an nmodel t h e n
( i n f a c t an mmodel whenever
5
m
5
n)
satisfying
has a model i f f i t has an nmodel f o r e v e r y $
dp($)
has no nmodel t h e n t h e r e i s a p r o o f o f
n i J,
. o f length
I+ln
2I'I
( t h e l e n g t h o f a p r o o f i s t h e number o f symbols i n i t ) .
4) If
J,
has an nmodel t h e n t h e r e i s no c u t
free proof o f 1 $
o f length
5
n.
Remark. We have n o t t r i e d t o m i n i m i z e t h e f u n c t i o n and numbers n o r we s h a l l do i t elsewhere i n t h i s a r t i c l e .
Proof: 1) a) I f of
J,
i s an nmodel o f
u, t h e n
fi[oydp(J,)li s
a
dp($)model
(check t h e d e f i n i t i o n ) .
2 ) Suppose
i s an nmodel o f J , . We d e f i n e by i n d u c t i o n on
a.
5
n,
A,
5 N,
On Logical Sentences in PA
147
such t h a t :
a ) l e t A, in
L
b) i f
At
appears in J , ) i s defined, 3 y (p(y,x) i s a subformula of
then f o r some b b
every non logical symbol
consist of a l l individual constants (w.1.o.g.
E
N2+1,
So clearly
IA,,
(we can forget
A
i s theunionof
and
E
ice’lsn’i= coCb,al;
“IA,I
a.
a
ji,
we demand t h a t there i s such
with a l l such b ’ s .
s Cnr.of individual constants in ~i I
+
”
,[Q,,tll I= ( 3 Y ) m ( Y i ) ,
c A e’
as the multiplication by
5
IJ,~
and
i s more than needed). As
models are non empty
Now l e t MI = Ne ? A
e’
and we can prove t h a t f o r every subformula
e(;)
of
JI, a c A R ’
na.
i C e s n l I= e [ a l
dp(e(i)) :
t
iff
i C e , n l I=
eG1
Me = M
, so
(just like Tarski Vaught c r i t e r i o n ) .
2) If
M
i s a model of
5
then l e t ( f o r a l l a. )
J,,
n> i s an nmodel of n If f o r every n <Me : R
<Me : .t
J,. 5
n > i s an nmodel of
<Me : R < w > which i s an wmodel of J,
.
Easily
3 ) Also immediate.
=
X ; ~  ~ , ~
[ $ ] ( m i ) dp(’)
nmodel of
J,,
X ,... ; , 3x nl , . . . , x n
and
e
J,,
Me
by compactness there i s
i s a model of
n
[There i s a quite short proof (of length
i s an
truth table of
lJ,T 1.
size 2 1 ~ 1 1
4 ) Just l i k e the proof t h a t every model s a t i s f i e s any provable sentence.
* * * * *
148
S.
Let
PA
SHELAH
be Peano a r i t h m e t i c , and
o f t h e i n d u c t i o n scheme o f l e n g t h
PAk
are included ( b u t except t h e instances o f
k
5
be Peano a r i t h m e t i c when o n l y i n s t a n c e s
t h e i n d u c t i o n scheme t h e r e a r e o n l y f i n i t e l y many axioms, which a r e included,hence i s finite).
PAk
Let
PAPL
be l i k e
PA, b u t wetake o n l y t h e i n s t a n c e s o f t h e
i n d u c t i o n h y p o t h e s i s w i t h no parameters, and and
PAk.
I t i s known
(Friedman t h e s i s , I t h i n k )
s i s t e n t . It i s c l e a r t h a t the consistency o f n
2
k,
m o n o t o n i c i t y we can t a k e o n l y
= <Mra..
.Q
5
n> where
If
rQ+l> r i ,
r Q> 1,
i.e. there i s
t h e axioms o f if
a r e equicon
n t k
,
PAFL has an nmodel" ( b y
k!=
r, < rl < r 2
22 k+n+'(r(i)+l),
F a s i n 3.2 and
N' >
a k  p l a c e f u n c t i o n s a t i s f y i n g 1 )  4 ) o f 3.2 f o r
N,
t h e n we can e x t e n d
N ' , i n one and o n l y one
way. __ P r o o f . By 3.2 ( 4 ) ( 3 ) (as i n t h e p r o o f o f 1.3 ( 1 ) ( b ) ) . 3.4.
Claim: I n
PA+PH
$* = ( V
r,
we can p r o v e (the
k, il)
r,
(N,
k , n )  p r i n c i p l e h o l d s f o r e.g.
,k+n+ f C r ( i ) + l l
).
N = 2 __ P r o o f : D e f i n e by i n d u c t i o n on
m
e,
i s s m a l l e r enough t h a n
putation). Applying
PH
and i n d i s c e r n i b l e f o r F ' ( A ~, 1
e,
a model
Am
4 Min C
157
On Logical Sentences in PA (we use an e q u i v a l e n t v a r i a n t o f P.H.).
: i
,
E
As i n 5 1
depends on t h e f i r s t v a r i a b l e o n l y . Now as i n t h e p r o o f o f 1.3
( 1 ) ( b ) we c o l l a p s e t h e s o l u t i o n below 3.5.
Claim: I n
PA+$*
N
.
we can p r o v e t h e c o n s i s t e n c y o f PA (hence
Proof. We can b u i l d a nonstandard model o f dard,
N
l a r g e enough, and
..., Ak)
F(A,,
mal code f o r which
1
( v y)
e a s i l y f o r subsequences o f
r
PA,+$*,
d e f i n e
F :
i s d e f i n e d as f o l l o w s :
let
/ = +
M, choose
cp(x,y)
3X[cp(X,y)
A
(v
z
< X)
and t h e n
( i f t h e r e i s one).
A,,
cp(x,y)
non s t a n 
be a f o r m u l a w i t h m i n i 
ic p ( Z , j ) l
(by the lexicographic order o f
E
k, n
"induction fails,i.e.
and t h e n t a k e minimal x
PAP$*).
1''
<Max
y.
y o , yl,
The r e s t i s as i n I 1
...>
)
the only
a d d i t i o n a l p o i n t i s why a r e a d d i t i o n and m u l t i p l i c a t i o n d e f i n a b l e ? T h i s i s by 3.1 ( 4 ) ( b ) .
5
4. On c o n s i s t e n c y s t r e n g t h
Let extending CON(T)
T PA
denote h e r e a ( r e c u r s i v e ) t h e o r y ( w i t h f i n i t e  s o r t , f i n i t e  l a n g u a g e ) b u t i t may a l s o "speak" on r e a l s and even a r b i t r a r y s e t s .
be t h e sentence ( i n
D e f i n i t i o n : We say
PA
T, scs T,
language) s a y i n g
T
i s consistent.
( t h e consistency strength o f
equal) than t h e consistency strength o f
T2)
if
PA
Let
t CON(T,)
T,
i s smaller (or f
CON(T,):
I t was observed t h a t e s s e n t i a l l y " l a r g e c a r d i n a l axioms a r e l i n e a r l y ordered"
(though i n some cases t h i s "has n o t y e t been proved").More e x a c t l y i t seems t h a t a l l s e t t h e o r i e s which has been c o n s i d e r e d so f a r , a r e l i n e a r l y o r d e r e d by Solovay
( I t h i n k ) has
found
T's
which a r e
5
incomparable,
s
cs b u t t h e y were
.
cs " p a r a d o x i a l " (i .e. have s e l f  r e f e r e n t i a l sentences). We s h a l l t r y t o g e t more r e a sonable ones.
* * * * * * * Let
PA+
be
PA t C O N ( P A ) .
We work i n s i d e
P A . A model w i l l mean one which
i s definable. Let
T,,
T,
be c o n s i s t e n t t h e o r i e s ( i n o u r " u n i v e r s e " which s a t i s f i e s PA).
s. SHELAH
158
T,
" s a y i n g " t h e r e i s a model o f
(think o f
PA+,
PA
+
CFMSI, o r o f course As T,
+
ZFC,
T, " s a y i n g " t h e r e i s a model o f
ATR, see Friedman, McAloon and Simpson
T,tCON(TI)
hence t h e r e i s MI
( 1 ) q Q ( n ) says
n
i s c o n s i s t e n t , hence ( b y Godel incompleteness has a model M,
M,
E
f ''$2(n)''
I=
such t h a t
(n
. By
t h e r e q u i r e m e n t on T,
i s a nonstandard i n t e g e r )
where
i s a n a t u r a l number and t h e r e i s a p r o o f o f s i z e
n
PA
ZFC+large c a r d i n a l s ) .
T, +CON(T,) + iCON(T, +CONTI)
iCON(T,),
n
of
t 1 CON ( Te)
PA
f ( 2 ) +,(n)
says
$
Q
(n)
but
f ( 3 ) T~ = PA + ( 3 n ) r$,(n) i s consistent with
i s t h e f i r s t such number.
n
As we have assumed t h a t
with
TI,
o r use
or
i s consistent,
theorem) M,
I rrlCAo;
T, says t h a t
has a model, c l e a r l y
+,~,,(2~")1 f
+ (Vm) l$,(m)
PA + 3 n$,(n)
( a s even
PA+
PA
i s consistent
.
PA+)
By a theorem o f Friedman (based on a n a l y z i n g Godel incompletness theorem) f
( 4 ) Tb = PA + ( 3 n ) C$,(n)
'
( 5 ) PA+
+
Ta, Tb
are
s
cs
incomparable.
L e t us p r o v e e.q.
*
Ta 1 CON(Ta)
rBecause f o r any model nition
11
PA
We s h a l l p r o v e t h a t csTb
217
+.
i s consistent with
Ta
$,(2
f
o f a mode:
phism from
No
Clearly
N,
of
N o of PA
(i.e.
PA++Ta, No
b e i n g a model o f
1: "N,
i n t o a p r o p e r i n i t i a l segment o f N,
satisfies
of bounded f o r m u l a s and
Ta PA,
has a d e f i 
i s a model o f P A " ) and an isomorN,.
as end e x t e n s i o n s p r e s e r v e t h e s a t i s f a c t i o n
f
$,(x),
PA+
$,(x)
a r e such f o r m u l a s . 1
Note a l s o t h a t (6)
PA
+
.
Ta I 7CON(Tb)
[Otherwise t h e r e i s a model a d e f i n i t i o n o f a model segment o f No I= T
a' formulas,
N, No
(i.e. f "$,(n)
and
f
N, o f No
No
of
PA+Ta+CON(Tb)
Tb a n d a n i s o m o r p h i s m g of
hence i n
No
No
there i s
onto a proper i n i t i a l
s a t i s f i e s t h e sentences s a y i n g t h o s e t h i n g s ) . AS 211 f and I $,(2 ) " f o r some n, b u t as $, $1 a r e bounded f commute w i t h e x p o n e n t i a t i o n , c l e a r l y N, I. "$,(g(n)) and
159
On Logical Sentences in PA 1 $,(22g(n))".
So
f 2m N, I= Tb hence f o r some m, N , I= "$,(m) and $ , ( 2 ) " . f f and $,(g(n))",hence by q 2 ' s definition g ( n ) , m have t o be
But
f
N , k "$,(m)
equal. B u t
N, k "$,(2
2m
)
and 7 $ 1 ( 2 ' g ( m ) ) " hence
g(n),
rn
should be unequal,
contraddictionl. By ( 5 ) and ( 6 ) clearly
PA+
+
C O N ( T a ) I$
(as PA+ + T~
CON(Tb)
is
consistent (by ( 3 ) ) ; t h i s implies
PA I f CON(Ta)
(7)
+
So Ta $ csTb.
CON(Tb)
. Tb $ csTa
i s t o t a l l y analogous. n f NOW (Woodin suggests) wecan replace the sentences ( 3n ) r $ , ( n ) +  1 $ ~ ( 2 ' ) I n f ( 3 n ) r $ , ( n ) + $ J 2 ' ) I by inequalities of the indicator functions corresponding t o
and
The proof of
f , , f,
T I and T, ( i . e . the function T , , T,).
t o the consistency of
exhibiting the
E.g. ( V n ) [ i f
f,(n)
ITsentences :
corresponding
i s defined then
so i s
f , ( f , ( f , ( n ) ) 1 and i t s negation. So i f we accept those functions as "mathematical"
( n o t j u s t reasonable methamathematical ) we get mathematical theories of incomparable consistency strength. (Originally we have used three t h e o r i e s ) . We a r r i v e t o the dangerous question o f which PA
function: on
see Paris and Harrington [ P H I ,
T's
have matheratical indicator
on many theories ( l i k e ZFC+large
cardinal) see Friedman rFl1 on ATR, see Friedman McAloon and Simpson rFMSl on 1
IT,CA,
see 5 2. Alternatively f o r an indicator function
f ( f * ( n ) + l ) , and use
$,
$3
where
$,
f
f*
define
= "the f i r s t
by
f*(O) = 0
n f o r which f ( n ) i s
:Q
f*(n+l)=
mod 4".
+ * * * * * * * Notice the following two phenomena
( A ) For any two natural s e t theories, not only they are
5
cs
comparable, b u t one
i s interpretable in the other ( o r expected t o be s o ) . ( 6 ) Similarly, f o r any theories, e.q. undecidab l i t y r e s u l t s are gotten by i n t e r 
pretation. Friedman rFr21 proved a theorem saying ( A ) i s really true. Concerning ( B ) however, in CSh1, (under C H )
the monadic theory of the re 1 order i s proven undecidable
without the usual interpretation. I n Gurevich and Shelah
CGSl1 t h i s
i s explained i t i s a Booleanvalued i n t e r p r e t a t i o n , and by CGS21 the usual i n t e r pretation i s impossible. Now we can t r a n s l a t e i t t o ( A ) : l i s t the reasonable axioms f o r the monadic theory of the real order (considered as a twosort model).
S. SHELAH
160
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161
CONTINUOUS TRUTH I Nonconstructive Objects Michael P . Fourman Department of Mathematics Department of Pure Mathematics Columbia University Uni vers i t y of Sydney New York, N . Y . 10027 N.S.W. 2006 U.S.A. Australia
W e g i v e a general theory of the l o g i c of p o t e n t i a l l y i n f i n i t e o b j e c t s , derived from a theory of meaning f o r statements concerning these o b j e c t s . The paper has two main p a r t s which may be read independently but a r e intended t o complement each o t h e r . The f i r s t p a r t i s e s s e n t i a l l y philosophical. In i t , we d i s c u s s the theory of meaning. We b e l i e v e t h a t even t h e s t a u n c h e s t r e a l i s t must view p o t e n t i a l i n f i n i t i e s o p e r a t i o n a l l y . The second p a r t i s formal. In i t , we consider t h e i n t e r p r e t a t i o n of l o g i c i n t h e gros topos of sheaves over t h e category of separable l o c a l e s equipped with t h e open cover topology. We show t h a t general p r i n c i p l e s of c o n t i n u i t y , l o c a l choice and l o c a l compactness hold f o r t h e s e models. We conclude with a b r i e f discussion of the philosophical s i g n i f i c a n c e of our formal r e s u l t s . They allow us t o reconc!le our explanation of meaning w i t h the "equivalence thesis , t h a t 'snow i s white i s t r u e ' i f f snow is white.
PROLEGOMENON Classical mathematics i s based on a p l a t o n i c view of mathematical o b j e c t s . The meanings of mathematical statements a r e determined t r u t h  f u n c t i o n a l l y . T h i s Fregean explanation of meaning j u s t i f i e s c l a s s i c a l l o g i c . The d e f i c i e n c i e s of such a view a r e amply discussed by Dummett C19781. A c o n s t r u c t i v e mathematician r e j e c t s t h e completed i n f i n i t i e s of classiGa1 mathematics. For h i m , t h e objects of mathematics a r e e s s e n t i a l l y f i n i t e . The meaning
of q u a n t i f i c a t i o n over i n f i n i t e domains is given o p e r a t i o n a l l y i n terms of a theory of c o n s t r u c t i o n s . T h e r e s u l t i n g l o g i c includes Heyting's p r e d i c a t e c a l culus and o t h e r p r i n c i p l e s ( e . g . choice p r i n c i p l e s ) .
As Dummett has s t r e s s e d , one t a s k of any philosophy of mathematics i s t o explain the a p p l i c a b i l i t y of mathematics. The p o t e n t i a l i n f i n i t i e s of experience exceed t h e f i n i t e o b j e c t s of t h e s t r i c t c o n s t r u c t i v i s t . They demanda mathematics of inf i n i t e objects. Naive a b s t r a c t i o n leads t o the i d e a l i n f i n i t e o b j e c t s of c l a s s i c a l mathematics. This i d e a l i s a t i o n has enjoyed remarkable success. However, the meaning of statements .of c l a s s i c a l mathematics remains problematic. Brouwer C19811 introduced t o mathematics p o t e n t i a l l y i n f i n i t e o b j e c t s such a s f r e e choice sequences. Consideration of t h e s e j u s t i f i e d , f o r Brouwer, i n t u i t i o n i s t i c l o g i c , including various choice and continuity princip2e.s. W e s h a l l consider a general notion of nonconstructive o b j e c t . For us, t o present such a notion i s t o give a theory of meaning f o r statements involving nonconstructive o b j e c t s .
Our nonconstructive o b j e c t s a r e not t h e p l a t o n i c ideal o b j e c t s of c l a s s i c a l mathematics nor t h e f i n i t a r y o b j e c t s of pure constructivism. They a r e p o t e n t i a l l y
M.P. FOURMAN
162
i n f i n i t e o b j e c t s r e l a t e d t o t h e l a w l e s s sequences o f K r e i s e l 119681and t o Brouwer's f r e e  c h o i c e sequences ( T r o e l s t r a 119771). The meanin s o f s t a t e m e n t s about t h e s e o b j e c t s cannot be g i v e n i n terms o f t r u t h c o n d i t i o n s ?as f o r c l a s s i c a l P l a t o n i s t mathematics) o r i n terms of c o n s t r u c t i o n s ( a s f o r n a i v e c o n s t r u c t i v i s m ) . The essence o f t h e s e n o n  c o n s t r u c t i v e o b j e c t s l i e s i n t h e i r i n f i n i t e c h a r a c t e r . They a r e n o t , i n g e n e r a l , t o t a l l y grasped. They a r e g i v e n i n terms o f p a r t i a l d a t a which may l a t e r be r e f i n e d . Meaning f o r statements a b o u t n o n  c o n s t r u c t i v e o b j e c t s i s g i v e n b y s a y i n g what d a t a j u s t i f i e s a g i v e n a s s e r t i o n .
To d e s c r i b e a p a r t i c u l a r n o t i o n o f n o n  c o n s t r u c t i v e o b j e c t i s t o d e s c r i b e t h e t y p e o f d a t a on which i t i s based. We c o n s i d e r v a r i o u s such n o t i o n s . Each c o n c e p t i o n o f d a t a g i v e s an e x p l a n a t i o n o f meaning w h i c h extends t h e range o f meaningful statements and may b e viewed as i n t r o d u c i n g new o b j e c t s i n t h a t i t a s c r i b e s meani n g t o new forms o f q u a n t i f i c a t i o n . I n f a c t f o r each t y p e o f d a t a we i n t r o d u c e a c o n c r e t e r e p r e s e n t a t i o n o f t h e n o n  c o n s t r u c t i v e o b j e c t s based on i t . Such a p r o j e c t i s n o t n o v e l : B e t h 119471 i n t r o d u c e d h i s models t o p r o v i d e j u s t such an e x p l a n a t i o n o f meaning f o r c h o i c e sequences. Our models g e n e r a l i s e Beth's. Dumnett 119771 makes a l e n g t h y c r i t i q u e o f t h e view t h a t t h e i n t e n d e d meanings o f o f t h e l o g i c a l c o n s t a n t s a r e f a i t h f u l l y r e p r e s e n t e d on B e t h t r e e s . Since o u r models g e n e r a l i s e B e t h ' s t h e y appear prima f a c i e t o be s u s c e p t i b l e t o t h e same c r i t i c i s m s . However, Dummett's remarks on t h e (non)consonance o f t h e i n t e n d e d meanings o f t h e c o n n e c t i v e s w i t h t h e i r i n t e r p r e t a t i o n i n B e t h t r e e s a r e d i r e c t e d a t a d i f f e r e n t problem f r o m t h e one we address. Dummett appears t o have o i e r l o o k e d t h e p o s s i b i l i t y o f s e p a r a t i n g t h e problem o f e x p l a i n i n g t h e c o n s t r u c t i v e meaning o f statements c o n c e r n i n g l a w l i k e o b j e c t s f r o m t h a t o f e x p l a i n i n g t h e i n t u i t i o n i s t i c meaning o f statements c o n c e r n i n g c h o i c e sequences. Although we know o f no s a t i s f a c t o r y e x p l a n a t i o n o f c o n s t r u c t i v e t r u t h ( i n p a r t i c u l a r , we agree w i t h Dummett t h a t B e t h models do n o t g i v e one), such a s e p a r a t i o n appears n a t u r a l . I t i s p o s s i b l e t o c o n c e i v e o f c o n s t r u c t i v e t r u t h i n d e p e n d e n t l y o f c h o i c e sequences. Given such a c o n c e p t i o n , Beth models p r o v i d e an account o f t h e i n t r o d u c t i o n o f n o n  l a w l i k e o b j e c t s . I t i s t h i s t y p e o f account we have g e n e r a l i s e d . By way o f example we now c o n s i d e r two n o t i o n s o f d a t a c l o s e l y r e l a t e d t o Beth models. They b o t h a r i s e f r o m t h e same i n f o r m a l p i c t u r e . The Imagine r e c e i v i n g f r o m Mars an i n f i n i t e sequence a o f n a t u r a l numbers. p i c t u r e i s o f a t i c k e r  t a p e which produces an i n d e f i n i t e l y c o n t i n u e d f i n i t e i n i t i a l segment a o f t h e sequence CL. (We w r i t e CL E a t o mean t h a t a i s an i n i t i a l segment o f a . ) We want t o examine t h e consequences o f t r e a t i n g such undetermined sequences s e r i o u s l y as sequences. ( L a t e r we s h a l l i n t r o d u c e more i n t e r e s t i n g examples )
.
A n a i v e view o f t h i s example c o n s i d e r s t h e stages b y which i n f o r m a t i o n a r i s e s : a t any stage, t h e p o s s i b l e f u t u r e d a t a i s r e p r e s e n t e d b y t h e c o l l e c t i o n N l t $ f o r a l l n alk @
163
E
N
P e r s i s t e n c e r e f l e c t s t h e i d e a t h a t knowledge, once j u s t i f i e d , i s secure. The i n d u c t i v e c l a u s e comes f r o m r e f l e c t i o n on t h e i n f i n i t e c h a r a c t e r o f a. Given a E a, t h e c o l l e c t i o n { a * I n c N I covers a l l p o s s i b i l i t i e s f o r f u t u r e data. I n general, i f we s t i p u l a t e b l k $ f o r b E B 5 N> X and a f u n c t i o n f: p t Z
E
WX E A.WZ E p t ZCpz = x As p t X i s r e p r e s e n t a b l e ,
+
such t h a t
$(x,f(z))l.
U I k V x 3 a o(x,a) iff
X i f f f o r some open c o v e r p: Z
zlk
ulk 3a
x
X
>
x
$ln2(nl,a)
U
$ln20~(nlo~,S)
f o r some 5 c A ( Z )
iff
Ult'Jz
E
z
$(P(Z),dZ)).
0
We do n o t know under what c o n d i t i o n s 6 descends t o g i v e a f u n c t i o n d e f i n e d on a c o v e r b y open s e t s . We can ensure t h i s b y c o n s i d e r i n g t h e open i n c l u s i o n t o p o l o g y on C i n w h i c h case we o b t a i n
1 Wx +
Wx
E
p t X.3a
E
3 open c o v e r Ui E
A.$(x,a) E
Ui.$(X'fi(X))
U(X) and f u n c t i o n s fi:
Ui
+
A such t h a t
.
We now c o n s i d e r c o n t i n u i t y . 4.4.
Proposition.
I f X,Y
a r e TI
then
1Vf:
pt
)#
+
p t W,
f is continuous.
M.P. FOURMAN
176
Proof. I f Uik f : p t )# + p t W t h e n f i s r e p r e s e n t e d b y 5 : X V E ( ) ( Y ) a b a s i c open o f W , w: W + U and x: W + X we have
Wkl iff
[S
0
(t;lw)(x)
E
<x,W>ll(v)
w It x
iff
1 regarding 5 V
O(Xx U) as an open o f
E
5 1(V) i s open.
Thus
Ulk
55
Iteration
E
U
+
Y in
(c.
For
v w
=
1 1 <x,w> 5 (V) =
iff
x
w
c51(v)lwl d e f i n e d a t U.
)#
0
We r e t u r n f o r a w h i l e t o c o n s i d e r a t i o n o f a general Grothendieck topos B = Sh(O,J). We c o n s i d e r t h e i n t e r n a l c a t e g o r y (I i n E g i v e n b y (E(U)
(c/u
w i t h r e s t r i c t i o n s g i v e n b y p u l l i n g back. [For those who w o r r y about coherence (one s h o u l d w o r r y ) , we remark t h a t a conc r e t e c a t e g o r y i n E w i t h an e q u i v a l e n t c a t e g o r y o f s e c t i o n s o v e r U i s g i v e n by c o n s i d e r i n g V / f t o be r e p r e s e n t e d as t h e element S o f (PV)(U) determined b y W / ~ V E S i f~f ~ f o v = g . So & i s an i n t e r n a l s m a l l f u l l subcategory o f E whose o b j e c t s a r e s u b f u n c t o r s o f representables.] We g i v e C_ a t o p o l o g y b y l e t t i n g
xi x
\/
Now f o r A
E
I E I we d e f i n e
w i t h r e s t r i c t i o n s f o r g: V
and f o r 5: Y/h
+
cover X / f i n
A, E +
X/g i n a/U,
+
i f Xi
+
X cover X i n
ShE(C,J) b y UkA_(X/f) A(X) U given b y r e s t r i c t i o n along f*g
by r e s t r i c t i o n along 5 Y
Any morphism A
&
B i n E induces
A
X
U +,B i n ShE($,J).
c.
Continuous Truth I
177
For those who p r e f e r g l o b a l d e s c r i p t i o n s , we associate t o A functors 6/U + E/U n a t u r a l i n U (i.e.
comnuting w i t h g* f o r g: V
+
E
If[ (pseudo)
U) as f o l l o w s :
where
For Y
'
,E/U
nf
U X
a, B
+
X
31
we have nh
P
.
npE whence
nhS* * nf
(as E.*
4ng)
U
and nhAy
* nPx (as
Ay)
E*Ax
.
This gives t h e r e q u i r e d arrow nhayA functor
+
What we o b t a i n i s an ( i n t e r n a l )
nfAXA.
C+EC
OP *
We s h a l l show t h a t t h i s preserves f i r s t order l o g i c . liere we work c o n c r e t e l y f o r t h e sake o f computations. A simple b u t more a b s t r a c t treatment w i l l appear i n Fourman and K e l l y C19831. We now consider a f i r s t  o r d e r language L w i t h s o r t s f o r t h e o b j e c t s o f E and operations symbols f o r i t s morphisms. I n f a c t t o avoid s i z e problems, we consider an a r b i t r a r y small f r a g ment o f such a language. We may consider L a l s o as a language i n K as a constant object ( v i a A ) . Working i n E we consider t h e i n t e r p r e t a t i o n o f L given by i n t e r p r e t i n g t h e s o r t A by A and each o p e r a t i o n f: A + B by t h e corresponding morphism 4 + &. 5.1
Lemma.
For f: X
+
U and g: X
~ l xk/ f k
9
+
V
iff
vlt
X/gl!
+
Ulk X/flk *g i s defined t o mean 0 f o r a l l g: X v As no r u l e decreases t h e complexity o f then IF i s closed under t h e r u l e s o f 9 we say assume t h a t t h e r e s u l t holds f o r subformulae o f 9.
Proof.
By i n d u c t i o n , i t s u f f i c e s t o show t h a t i f
v ~ XF/ g l k
+
.
Only (+)+ and ( W ) ' present any d i f f i c u l t i e s . r e s u l t f o r @ and $.
Me consider (+)+, and suppose t h e
Suppose t h a t f o r a l l E: W U and a l l h: Z + g*X, i f W Z/(E*f h) Ip*01(f*E 0 h) Then i f n: W ' + V and h ' : Z + rr*X a r e such then W l k Z / ( E * f h ) I p d ( f * E  h ) . that W ' Z ' / ( n * g h ' ) IF @ l ( g * n h ' ) then by i n d u c t i o n hypothesis +
0
0
M.P. FOURMAN
178
It
U k  Z ' ( f 0 g*no h ' ) * $1(g*no h ' ) whence ( l e t t i n g 5 = i d and h = g*n h ' ) we have 0
U
Z'/(
f
0
g*n
0
It * $1 (g*n
h'
h') So V l k X / g / k ~  t $ . The p r o o f
i n particularW'IkZ'(n*goh')lkJil(g*qoh').
0
I
for V 5.2
i s similar.
Theorem.
Proof.
0
F o r Q a f o r m u l a o f L w i t h a p p r o p r i a t e parameters
U IF'' X/flk Q" i f f xlk Q . F i r s t l y , t h i s i s w e l l formed: Parameters f o r Q a t X / f a r e elements o f which a r e g i v e n as elements o f A(X) and a r e t h u s parameters f o r $ a t X.
m)
We proceed by i n d u c t i o n .
T h a t i s , we show t h a t i f we d e f i n e
It* i n t e r n a l l y
It
Ulk X / f * Q iff X l t Q c l o s e d under t h e d e f i n i n g c l a u s e s o f l k i n t e r n a l l y , (whence UIk X / f 1 1@ X\k $) and i f we d e f i n e \I by + X $ i f f Ulc X / f Ik @ t h e n i s c l o s e d under t h e d e f i n i n g c l a u s e s o f (whence Xlk Q *VIE X / f l t  Q). then
by

it* i s
It+
\kt
As t h e o p e r a t i o n s A + B a r e j u s t t h o s e i n h e r i t e d f r o m E, terms a r e i n t e r p r e t e d a l i k e i n b o t h contgxts: Thus i f [ T I = Uo] t h e n UlkUrl = Dull, so i s closed under ( = ) + and i f Ulk U ~ l l= Uol t h e n UIk T = a,
11'
so
IF*
i s c l o s e d under ( = ) +
It and \I* a r e c l o s e d under ( A ) ' , (v)', (3.)' i s t r i v i a l . F o r I, suppose 1 1 ' $Ifi f o r fi: Xi X i n some cover o f X t h e n X I 1 Xi/fi $Ifi and by I i n t e r n a l l y Xik X / i d l k $. I n t h e c o n t r a r y d i r e c t i o n , suppose Ulk Xi/g fi IF* $Ifi f o r some c o v e r of X as above. Then Xi $Ifi so Xlk Q t h a t i s Ulk X/g Q. F o r (+)+, f i r s t suppose t h a t f o r a l l f: V U i f V I   + ~ lt hfe n V I k + ~ l f Then . we c l a i m U I U / i d l k @ + I$,because f o r a l l g: W + U and a l l h : V + W , i f W @1g h, t h e n V It+$ l g h so V IF $19 h, t h a t i s W V/h v/h Jilg h. Conversely, iff o r a l l g: W + U and a l l h: Z + g*X, where f: X + U, i f WIE Z / g * f h \I* $ l f * g h , t h e n X I k @ + $, because f o r h: Z X i f Z \ k $ l h then U l t Z / f h It* $ l h so Ulk Z / f h It*Jl?h which g i v e s Zlk $Ih, so Ulk X / f I/* Q Ji. That
+
Xi
+
0
+
IF
0
o
0
0
+
t
0
0
0
+
0
The p r o o f f o r W+ i s s i m i l a r .
0
We view t h i s thorem as a s s e r t i n g t h a t i n t h e topos E t h e n a i v e n o t i o n o f t r u t h g i v e n by t h e e q u i v a l e n c e t h e s i s i s consonant w i t h t h e t h e o r y o f meaning g i v e n b y t h e n o t i o n o f f o r c i n g o v e r t h e s i t e &. O f course t h i s may seem vacuous as i t appears t h a t B i s manufactured w i t h t h i s r e s u l t i n mind. However, i n t h e case o f p r i m a r y i n t e r k t f o r t h i s paper, t h e r e s u l t s o f 84 a l l o w us t o r e g a r d (I i n t e r n a l l y as a f u l l subcategory o f Loc(E) equipped w i t h t h e open cover t o p o l o g y . I n f a c t , i f Q i s t h e c a t e g o r y o f s e p a r a b l e l o c a l e s , we may i d e n t i f y (I as a c a t e g o r y o f s e p g r a b l e l o c a l e s i n E. We s h a l l deal w i t h t h i s , among o t h e r t h i n g s , i n a sequel t o t h i s paper. Given f: X
f
U we may view an element a o f A(X) as a f u n c t i o n : U
It a:
X/f
+
A,.
T h i s a l l o w s us t o r e p h r a s e o u r theorem. 5.3
Corollary.
ulkX/flk $(a)
iff
Ulk~tE
X/f@(a[t)).
0
We view t h i s as a g e n e r a l f o r m o f t h e e l i m i n a t i o n theorem ( c f . T r o e l s t r a C19771
Continuous Truth I
179
The appropriate theory o f continuous t r u t h CT has an axiom f o r each pp.33,79). clause i n t h e d e f i n i t i o n o f X/f/k$(a). For example, t h e clause f o r 3 gives the axiom o f l o c a l choice Y t E 3 y $ ( a ( t ) , y ) i f f 3 open cover p: Z >> X and continuThe t r a n s l a t i o n T $ o f a formula $ w i t h ous f: Z + Y such t h a t W z $ ( a ( p ( z ) ) , f ( z ) ) . o u t f r e e lawless v a r i a b l e s i s given by T$ :def/k $.
X
CODA A general n o t i o n o f nonconstructive o b j e c t i s given by i n t e r p r e t a t i o n s i n Grothendieck t o p o i . The process o f i t e r a t i o n described i n 55 shows how we may view ( i n t e r n a l ) t r u t h i n t h i s i n t e r p r e t a t i o n as given by a nonstandard theory o f meaning. The clauses d e f i n i n g t h i s g i v e axioms f o r the corresponding theory o f continuous t r u t h CT and an " e l i m i n a t i o n " t r a n s l a t i o n . By construction, CT tf T$ and f o r formulae i n t h e l a w l i k e p a r t o f t h e language T $ 5 $. The p r o o f t h e o r e t i c content o f t h e e l i m i n a t i o n ;
I$
CT
$
iff
ID
T$,
requires f o r m a l i s a t i o n o f our treatment i n an appropriate theory I D o f i n d u c t i v e d e f i n i t i o n s . We do n o t undertake t h i s here. A f i n a l example o f an u n f i n i s h e d o b j e c t i s t h i s paper. Some o f t h e r e s u l t s , i n p a r t i c u l a r c o n t i n u i t y p r i n c i p l e s i n sheaves over s i t e s , go back t o 1978 and were much i n f l u e n c e d by discussions w i t h S c o t t and Hyland. Some r e s u l t s are s t i l l being r e f i n e d . Other p e r s i s t e n t i n f l u e n c e s have been those o f Joyal and Lawvere on t h e one hand and o f K r e i s e l , T r o e l s t r a and Dummett on the other. This research has been supported a t various times by the N.S.F. (U.S.A.), the S.R.C. (Netherlands), and t h e A.R.G.S. ( A u s t r a l i a ) , and made e a s i e r (U.K.), t h e Z.W.O. by t h e h o s p i t a l i t y o f many people n o t a b l y C h r i s t i n e Fox, I r e n e Scott, Karen Green, and Imogen K e l l y . I am g r a t e f u l .
REFERENCES A r t i n , M., Grothendieck, A., Verdier, J.L., ThGorie des Topos e t Cohomologie, E t a l e des Sch6mas (SGA4), (Lecture Notes i n Math. 269, 270, SpringerVerlag, B e r l i n , 1972). Beth, E.W., Semantical Considerations on I n t u i t i o n i s t i c Logic, Indag. Math., 9(1947), p.5727. Boileau, Andr6 & Joyal, Andr6, La logique des topos, J.S.L.
46(1981), p.616.
Brouwer, L.E.J., Cambridge Lectures on I n t u i t i o n i s m , D. van Dalen, ed. (Cambridge U n i v e r s i t y Press, 1981). Dummett, Michael, Elements o f I n t u i t i o n i s m , (Oxford U n i v e r s i t y Press, 1977). Dummett, Michael, T r u t h and
o t h e r enigmas, (Duckworth, London, 1978).
Fourman, Michael P., The l o g i c o f Topoi, i n Handbook o f Math. Logic (ed. Barwise, J.), (NorthHolland, 1977), p.105390.Fourman, Michael P., Notions o f Choice Sequence, Proc. Brouwer Symposium, (ed. T r o e l s t r a , A. and van Dalen, D.), (NorthHolland, 1982). Fourman, Michael P. & Grayson, Robin J., Formal Spaces, Proc. Brouwer Symposium, (ed. T r o e l s t r a , A. and van Dalen, D.), (NorthHolland, 1982). Fourman, Michael P.,
T1 spaces over t o p o l o g i c a l s i t e s , JPAA,
( t o appear), 1983.
180
M.P. FOURMAN
Freyd, P e t e r , Aspects of Topoi, Bull. A u s t r a l . Math. SOC., 7(1972), p.176. I s b e l l , John, Atomless p a r t s of spaces, Math. Scand., 31(1972), p.532. Johnstone, P e t e r T . , Topos Theory, (Acad. Press, London, 1977). Johnstone, Peter T . , Stone spaces, (Acad. Press, London, 1982). J o y a l , Andre, & Tierney, Myles, An extension of the Galois theory of Grothendieck, p r e p r i n t , 1982. Kreise!, Georg, Lawless sequences o f natural numbers. p .22248.
Comp. Math. 20(1968),
Makkai , Michael & Reyes, Gonzalo, FirstOrder Categorical Logic, (Lecture Notes in Math. 611, SpringerVerlag, 1977). Moschovakis, Joan R., A topological i n t e r p r e t a t i o n o f secondorder i n t u i t i o n i s t i c a r i t h m e t i c , Comp. Math., ( 3 ) , 26( 1973), p.26175. S c o t t , Dana S., Extending t h e topological i n t e r p r e t a t i o n t o i n t u i t i o n i s t i c a n a l y s i s , Comp. Math. 20(1968), 22248. S c o t t , Dana S . , I d e n t i t y and Existence i n I n t u i t i o n i s t i c Logic, Proc. Durham Symposium, (ed. Fourman e t a l . ) (Lecture Notes i n Math. 753, SpringerVerlag, 1978) , p. 66096. T r o e l s t r a , Anne S . , Choice Sequences, (Oxford University P r e s s , 1977). Wraith, Gavin C . , Lectures on elementary t o p o i , Model theory and t o p o i , (ed. Lawvere F.W. e t a l . ) , (Lecture Notes i n Math. 445, SpringerVerlag. B e r l i n , 1975), p. 114206. Wright, Crispin, W i t t g e n s t e i n ' s Philosophy of Mathematics, (Duckworth, 1981).
LOGIC COLLOQUIUM '82 G. Lolli, G. Longo and A . Marcia (editors) 0 Elsevier Science Publishers B. V. (NorthHolland), 1984
181
HEYTINGVALUED SEMANTICS R.J. Grayson
*
Institut fur mathematische Logik und Grundlagenforschung Einsteinstrafle 6 4 ,
4400 Munster, West Germany
Introduction. Chapter I.
The Logic o f HSets.
5 1 . Complete Heyting algebras. § 2. Interpretations of propositional logic
5 5 9
3. Hsets. 4.
Interpretations of predicate logic.
5 . Number systems.
§ 6. Complete Hsets.
5
7. Interpretations o f higherorder logic.
Chapter 11. Mathematics in HSets. § 8.
5
Some internal constructions.
9. Internal topologies.
§ l0.Choice principles.
9
11.Continuity principles.
References
Introduction. In this paper we develop a semantics for intuitionistic systems in which sentences are given "truthvalues'' in complete Heyting algebras (cHa), just as sentences of classical set theory are given values in complete Boolean algebras ([MD], for example). T h e use o f the lattice of open subsets of a topological space t o interpret intuitionistic propositional logic goes back t o Tarski ([Ta,RS]). Extensions t o predicate logic were made b y Beth and Kripke (ID]) and applied t o metamathematical results for arithmetic by Smorynski ([Tr]). Further interest was drawn t o the area by the topological interpretations of analysis in [Sl,Mo,VD], where it was shown that "Brouwer's Theorem", on the continuity o f all functions between reals o r the Baire space, could be modelled in this way. In addition, Bishop's book ([Bi]) showed the feasibility of constructivism and gave new impetus t o the investigation of constructive and intuitionistic systems. A t the same time, interest has arisen from the theory of topoi,
*
Research Fellow of the AlexandervonHumboldtFoundation
R.J. GRAYSON
182
which can be seen as a categorytheoretical formulation of intuitionistic higherorder logic ( [ F l l , for example). Other kinds of semantics are also suggested by this approach, for example, sheaves over sites ([MR]). However, the level of generality of Heytingvalued semantics seems to provide a natural stoppingpoint: the notion of cHa is simply an algebraicisation o f the notion of "truthvalue" for intuitionistic predicate logic, staying within the conceptual framework of topological, Beth and Kripke models. The general theory of sheaves over a cHa (here called Hsets) is worked out in great detail in [ F S ] , where it is shown how they model intuitionistic higherorder logic (the extension to set theory is made in [Gl]). This paper is designed as a selfcontained introductory exposition of the basic definitions and results, which it is hoped will enable the interested reader then to come to grips with more detailed treatments as well as with more specialised papers in this area. The paper falls into two chapters. In Chapter I we describe successively the interpretations of propositional, predicate and higherorder logic over a cHa. In Chapter I1 we develop some analysis and topology in these models, with particular emphasis on topological models and on the interpretation of various principles of choice and continuity. We close with Joyal's very elegant proof, using topological models, of a derived rule of local continuous choice for intuitionistic higherorder logic. I have not attempted on the whole to assign credit too exactly, beyond references to the literature, but I should like to acknowledge here the contributions of Dana Scott, whose influence on the whole treatment should be clear, and of Mike Fourman and Martin Hyland, who have stimulated my interest in the subject over the years. I thank the AlexandervonHumboldtFoundation, Bonn, for financial support, and the Institut fur mathematische Logik und Grundlagenforschung, Miinster, for their hospitality.
CHAPTER I.
5
THE LOGIC OF HSETS
1 . COMPLETE HEYTING ALGEBRAS
We begin by defining the structures which are to act as our domains of "truthvalues". Although we will be mostly concerned with topological examples, this more general, algebraic setting seems to make the essential features clearer, besides providing further examples (see 9 . 7 for example). Much information on the classical theory of complete Heyting algebras (cHa) may be found in [ R S ] and on the constructive theory in [ F S , Chapter I]; for we want to be handle our models "constructively" too (see 7.8 for further discussion of this point). 1 . 1 Definition. A complete Heyting algebra
lattice (H,Z), with finitary and infinitary by h , l \ , v , V ,
is a complete
meet
and join denoted
satisfying the distributive law, for pEH and ASH, phVA
E
V(phq1qEA).
Hereafter H will always denote a cHa, with elements p,q,... also the notation T for VH. the "top" element, and 1 for "bottom" one.
AH,
.
We use
the
HeytingValued Semantics
Logically, the order relation tion. 
5
183
is read as the relation of implica
In addition one may define in any complete lattice an
implication operation by
5
(p9) = V I r t p A r
q}.
AS a special case we have negation ~p defined as (p+I), which equals V{r I pAr=I) 1.2 Lemma. In any cHa H the implication operator is characterised by the adjunction rZ(p+q)
iff
(phr)lq.
Proof. If ( p n r ) ~ q ,then rL(p+q) always holds, by definition of implication. If H is a cHa and rl(p+q), then the distributive law gives pAr
5 =
PAVCSIPAS5 V{pAS [ P A S
q}
5 qj
5 4.
Proof. (i)

(iii) follow at once from Lemma 1 . 2 .
Since ~p(~p,
(iii) gives ph7p=l and then p5,.p. From (i) and qATq=l we obtain ph(p+q)h~q=I, hence applying 1 . 2
(p+q)
5
~ ( p h ~ qand ) (ptq1Aq 5 TP, by
again gives (p+q)
5
(iii)i
(q+p). The remainder is left as
an exercise. 1.4
Examples. a) The open subsets O(T) of any topological space T
form a cHa under inclusion, 5 . A , V , V are the settheoretic fl,U,u while hA=Int ( O A ) and T
(U+V) = IntftI tEU + tEV).
is T , I is the empty set
In this context we use u , V ,
d , and
...
rU is Int(T'U1.
for elements of O ( T ) , and s , t , . . .
for elements of T. We call such cHa topological; ways of obtaining
R.J. GRAYSON
184
nontopological examples may be found in [FS,S2]. b) As special cases of topological cHa we have those arising from partial orders ( K , c ) , where K is given the topology of upwards closed subsets (that i s , O ( K ) consists of those P such that Vi,jEK.j)iEP + j E P ) . This provides the connection between Kripkemodels based on partial orders and semantics with "truthvalues" in topological cHa (see 3 . 3 (c))
.
c) For a similar connection with Bethmodels based on a partial order ( K , Z ) , one takes T to consist of all maximal chains a in K , with O(T) having as subbasis the sets {UlfEa) for iEK. 1.5 Heyting Algebras. A lattice equipped with an implication having the property of Lemma 1.2 we may call simply a Heyting algebra. These are treated in [RSI under the name of "relatively pseudocomplemented" lattices; it is shown there that all such lattices satisfy & t Jfinitary distributive laws, as well as the infinitary one for such joins as exist.
For the purposes of 2.5 it is useful to note the following simple completion process for any Heyting algebra H: Let O(H) be the topology of downwards closed subsets of H (compare 1 . 4 (b)), and let J be the Joperator ( I F S , 2 . 1 1 ] ) defined by J ( U ) = the set of all joins of subsets of U which exist in H. Then p W [PI = {qlqEx).
The interpretation of this term as an Hset, according to 7.6, is then the exponent of A and A B , which consists of all relations R,S on AaXAB with [[ E(R) 11
=
[[ R a total functional relation]]
Ba
and [[ R=Sl]
=
[[ E(R)II
Ba
aa
A
{ [ [ R(a,b)ll
aEA,,
*+
[[ S(a,btll 1
b€AB)
7.8 Now we ask the reader to look back over this first chapter and see that the definitions of Hsets and validity in them, and the proof o'f soundness of standard interpretations, can all be carried out within the system IHLN of 7 . 3 . This means, for example, that we can iterate the construction of the models inside any universe of Hsets, just as forcing is iterated in classical set theory (see [FS,§9] for example). More interestingly, perhaps, we can use the provable soundness of the interpretations to obtain derived rules for the system. We give an example of this, due to Joyal, in 1 1 . 5 ; other examples may be found in [Be,H2,FJ]. So in Chapter I1 we will be concerned to note, as we did in 5, what principles are needed to prove the validity of various assertions in various models, arguing so far as possible "constructively", i.e. within the system we are interpreting. In order to distinguish what is assumed to hold "on the outside" (or "in the ground model") from that which is valid in the interpretations, we use the terms external and internal.
The above cor.siderations all extend mutatis mutandis to systems of with the powerset axiom and full comprehension, as formulated in [Gl] and exploited in [HZ]. The general problem of interpreting a set theory, with only the axiom of exponents, within such a theory is dealt with in [G3]; applications of this are made in [Be].
set theory ~
CHAPTER 11.
5 8.
MATHEMATICS IN HSETS
SOME INTERNAL CONSTRUCTIONS
We are now ready to interpret constructions within the system IHLN of higherorder logic with a sort for natural members (7.3) in Hsets. The integers, rationals, real numbers, functions etc. appear as types (7.5) in this language, which we want to interpret as Hsets according to 7 . 6 . Such characterisations will generally be
HeytingValued Semantics
197
only "up to isomorphism", in the following sense. 8.1 Definition. An isomorphism between Hsets A and B is a total functional relation ( 4 . 7 ) on AxB which is internally oneone and onto. As in 4.8 and 6 . 3 , in particular cases an isomorphism may be given by a function from A to B which is internally oneone and onto, or even (for example, when both A and B are complete) by a pair of functions F:A+B and G:B+A which are inverse to one another:
VaEA.
[[ Ea]]
VbEB. [[ Eb]]
and
5 5
[[ a=G(Fa) 11 [[ b=F(Gb) 11
.
The extension to isomorphisms of structures is made in the obvious way. As in classical mathematics there is only one structure (up to isomorphism) satisfying Peano's axioms for arithmetic in any standard interpretation fn Hsets. We do not prove this fact but only show that the Hset N with constant structure ( 5 . 1 ) does satisfy the axioms, and hence can serve as the interpretation of the sort N. 8 . 2 Proposition. f j with the standard successor function S satisfies Peano's axioms for arithmetic, including induction in the form
VXEP(2). O E X A v x E X . SXEX
+
VXEN. XEX
Proof. Since the interpretation of firstorder sentences is always absolute (5.11, the firstorder axioms are trivial. To prove inguction one shows, by an external induction, for any predicate P on N. that OEP
if
q =
then
VnEN. q
By definition q ( " of 4.4,
5
5
5
:
95" VxEP.SxEP11,
[[ nEP11
A
[[ nEP11, q
VXEP. SXEP]], nEP11
OEPII and VnEN. q
hence, if q
A
5
that is, as in the proof
[[ SnEPl];
[[ SnEPl].
8.3 We leave the reader to check that, for some standard definitions of the integers and rationals as types obtained from products of N, the corresponding Hsets, according to 7.6, are isomorphic to the constant Hsets 2 and Alternatively 5 and with constant structure (5.1), can be shown to be the unique Hsets (up to isomorphism) with certain properties (e.9. 6 is a countable dense linear order without endpoints).
a.
0,
We are now ready to formulate and prove the "converse" of Theorem 5.6, giving a characterisation of real numbers in topological models. 8.4 Definition. A Dedekind cut (in the rationals) is a pair (L,U) of subsets of Q which are inhabited, disjoint, closed downwards (resp. upwards), open (in Q), and close together: that is, (i) (ii) (iii) (iv) (V)
3pEL A 3pEU L n u = @ (pqEU + pEU) (PEL + 3qEL. q>p) A (PEU + 3qEU. q
(i) ~ (ii)
E
~ = +a a = a
be a c o m b i n a t o r y model.
~ A V X E(ax) =
vx l . . . x i
~9
(iii) l i s stable
Proof.
Ea a =
~
...x . )
€(axl
ax. = ax l . . . x . ,
O . Thus ( B * b ) i s c l e a r l y r e l a t e d t o t h e type k + { i ~ l .This w i l l be used in several places i n t h i s paper and more formally s t u d i e d , p a r t i c u l a r l y when dealing with D, models. In t h e present paper t h e notion of types and t h e i r i n c l u s i o n p r o p e r t i e s a r e a b s t r a c t l y formalized i n t h e d e f i n i t i o n of Extended Abstract Type S t r u c t u r e (EATS). Actually EATS a r e information systems i n t h e sense of S c o t t [211 ( c f . 1 . 9 ) . I t i s i n t e r e s t i n s t o consider EATS given over a p p l i c a t i v e s t r u c t u r e s (they a r e c a l l e d ETS). ETS can be viewed a s i n t e r p r e t a t i o n s ( i n t h e sense of S c o t t ) of formal types. Continuous a p p l i c a t i v e s t r u c t u r e s over EATS ( f i l t e r domains) a r e defined. We then i n v e s t i g a t e embeddings and isomorphisms between f i l t e r domains. In p a r t i c u l a r , t h e f i l t e r domain defined in [21 i s shown t o be " u n i v e r s a l " in t h a t any f i l t e r domain i s isomorphic t o the range of a c l o s u r e operation which i s an element of IFI. Moreover we o b t a i n simple r e l a t i o n s between t h e p r o p e r t i e s of "2 of an EATS and the c l a s s of r e p r e s e n t a b l e f u n c t i o n s over the a s s o c i a t e d f i l t e r domain ( c f . 2.13). Some f i l t e r domains can a c t u a l l y be turned i n t o models of type f r e e 1calculus ( f i l t e r Amodels). An i n t e r e s t i n g c l a s s of them (which has a simple c h a r a c t e r i z a t i o n in terms of " < " ) i s t h e c l a s s of f i l t e r domains i n which a l l continuous functions a r e representable. However t h e r e e x i s t a l s o f i l t e r Amodels i n which not a l l continuous f u n c t i o n s a r e r e p r e s e n t a b l e ( s e e 4.11). Embeddings and isomorphisms between r e f l e x i v e domains ( c f . [ l n ) and f i l t e r h o d e l s a r e s t u d i e d : i n p a r t i c u l a r any Dmspace i s isomorphic t o a ( s u i t a b l e ) f i l t e r h o d e l . L a s t l y j u s t using ( e a s i l y axiomatizable) a b s t r a c t 'I , where X i s a set,weX,"A"and 14diare t o t a l f u n c t i o n s f r o m X x X t o X and k " i s a p r e o r d e r r e l a t i o n on X s a t i s f y i n g : 1. a L W 2. w 5 w+w 3. a L a A a 4. h b l a aAb5b 5. (a;b) A +c) L a +(bhf) 6. a s . , b d * a h b c a ' h b 7. a ' z a , & ' * a + b c a ' + b ' e X. where a,b,c,a',b'
(7
L e t a ?.b i f f a5bQ. Observe t h a t w w + w
and a
A
(bAC)
(ahb)hc.
...,in}, then
NOTATION: I and J w i l l always be f i n i t e s e t s o f i n d i c e s . I f I= {il, ,A a . means a . A a
I
'
11
i2/\
i;
1.2. EXAMPLES. ( i ) L e t T be t h e s e t o f f o r m a l t y p e s b u i l t f r o m w a n d a ( c o u n t a b l e ) s e t At= {$o, 4 l,. 1 o f t y p e v a r i a b l e s by t h e ( s y n t a c t i c ) o p e r a t o r s
..
" +'I
and
"A"
o f 1.1 t h e n
o f type formation. F=is
I f ''zO" i s t h e minimal p r e o r d e r s a t i s f y i n g 17 t h e f r e e EATS o v e r g e n e r a t o r s
...
4 0 . 4 ~ ~
defined i n [ 21. (ii)Consider $ = where P i s t h e s e t o f w . f . f . of ( p r o p o s i t i o n a l ) d e r i v a t i v e + A  l o g i c [16, p. 2851 , aeP and "(I' i s d e f i n e d by " p l q i f f p t q " . 9' i s an EATS s i n c e 17 t r i v i a l l y h o l d . and
Our " c o n c r e t e " EATS w i l l always be g i v e n o v e r an a p p l i c a t i v e s t r u c t u r e and " A " w i l l be ( i n t e r p r e t e d b y ) s e t t h e o r e t i c i n c l u s i o n and i n t e r s e c t i o n .
"2'
1.3. D e f i n i t i o n . ( i ) L e t be a ( p a r t i a l ) a p p l i c a t i v e s t r u c t u r e and A , B g . Define then A+B = { d r 01 VeeA d.eeB 1 €PO (if i s a p a r t i a l o p e r a t i o n , b y d.eeB we mean: I'de i s d e f i n e d and d e B " ) . ( i i ) L e t be an a p p l i c a t i v e s t r u c t u r e . An Extended T e S t r u c t u r e (ETS) ( o v e r D ) i s an EATS S = < p, 5 , n, +,D>, where PSPD, C_ a n d K p a r e s e t i n c l u s i o n and s e t i n t e r s e c t i o n and B t P .
"."
I n o t h e r words an ETS i s a s e t o f subsets o f an a p p l i c a t i v e s t r u c t u r e , n o t c o n t a i n i n g t h e empty s e t , and c l o s e d under " n " and ' I + ' ' . I t i s t h e n easy t o check t h a t t h e c o n d i t i o n s o f 1.1 a r e s a t i s f i e d (indeed, "L'" i s a p a r t i a l order). 1.4. EXAMPLE. ( i ) L e t Definition. ( i ) L e t Ss < X , ( , i s a f u n c t i o n V: T + X such t h a t :
be
an
EATS.
Then
a
2. V( O A T ) = V ( U) A V ( T) 3. v(U*T)= v ( U)* V(T). We say t h a t < S,V> i s a t e model (a c o n c r e t e t y p e model when S i s an ETS). (ii)If<S.V> i s a t y p h t s t h e o r y TV i s g i v e n by TV = { O T I V ( O ) G v(T)}*
E.VT
stands f o r
U ~ T
Tv.
N o t i c e t h a t g i v e n any EATS S, we can always f i n d many V:T+X such t h a t <S,V> i s a t y p e model. C l e a r l y , i f X i s c o u n t a b l e , V can be made s u r j e c t i v e . O f course, t h e c o n d i t i o n on c o u n t a b i l i t y may be dropped i f one t a k e s t h e s e t A t o f atoms o f the desired cardinality. F i.n a l l y , i f V i s o n t o , Obviously Tv= T (Tv) and ToETV ( i . e . ''2; extends " 3 " ) one c l e a r l y has
< X,L,
I
A
+ ,
@>
.
Some more work can be done w i t h EATS, l o o k i n g a t c o l l e c t i o n s o f t h e i r subsets. be an EATS. D e f i n i t i o n . L e t S= <X, < A , *,w> An a b s t r a c t f i l t e r x o f 3 i s a non empty subset o f X such t h a t : 1. a.be x =. h b e x 2. a; x, a 3 * k x . ( i i ) If AGX, t A i s t h e a b s t r a c t f i l t e r generated by A. I f A= {a), f o r fIa1. ( i i i ) IS1 i s t h e s e t o f a b s t r a c t f i l t e r s o f S ( f i l t e r domain o f S ) . 1.7. (i)
I f S i s an ETS, IS1
i s clearly the set o f f i l t e r s o f
f a stands
S i n t h e u s u a l sense.
1.8. LEMMA. < IS1 & > i s a complete a l g e b r a i c l a t t i c e , where f w and X a r e t h e l e a s t and t h e l a r g e s t elements ( r e s p e c t i v e l y ) . Moreover i f x,ye I S I : (i) x w = t (xL5) ( i i ) xny = X ~ Y ( i i i ) IfA G l S l i s a d i r e c t e d s e t , t h e n U A = U A . ( i v ) The finite elements are exactly the principal filters, i.e. x = u { f a 1 facx 1 Proof. Easy. 0 1.9. REMARKS. (i) EATS a r e i n f o r m a t i o n systems i n t h e sense o f S c o t t 1211. I n f a c t , an EATS s<X,< , A , *, w> i s an i n f o r m a t i o n system (X,u,Con,+) where Con c o n s i s t s o f a l l  f i n i t e subsets o f X and, i f A = {al, a n } , Atb iff a,&
... ~a~
...,
5 b (and +Ib i f f
w~
b ) . Moreover
IS1 i s t h e s e t o f elements o f
t h e c o r r e s p o n d i n g i n f o r m a t i o n system. ( i i ) Any ETS < p , c,n , *,D > i s a neighbourhood system i n t h e sense o f [ Z O I . Moreover i f we d e f i n e : AfdB * deA+B (where deD and A , W P ) t h e n f d i s an approximable mapping, as d e f i n e d i n [201.
245
Extended Type Structures and Filter Lambda Models FILTER DOMAINS
2.
This s e c t i o n mainly deals w i t h p r o p e r t i e s o f f i l t e r domains ( o f EATS), viewed as a p p l i c a t i v e s t r u c t u r e s . I n t h e sequel complete l a t t i c e s w i l l always be considered w i t h t h e S c o t t topology ( c f . [17]). D e f i n i t i o n . ( i ) I f D i s a complete l a t t i c e ( w i t h respect t o "I")and " ' " : DxD+D i s continuous, then is a continuous a p p l i c a t i v e s t r u c t u r e . ( i i ) A continuous a p p l i c a t i v e s t r u c t u r e i s a l g e b r a i c i f f D i s algebraic.
2.1.
Given any EATS structure. 2.2.
S, one may t u r n Is1 i n t o an a l g e b r a i c continuous a p p l i c a t i v e
D e f i n i t i o n . L e t S be an EATS. Define: x IS1 + IS1 by xy = { b 13aEy a + bexl.
"'":ISI
2.3. (ii)
LEMMA. ( i ) x , y r I S I * x ' y ~Is! i s an a l g e b r a i c continuous a p p l i c a t i v e s t r u c t u r e .
Proof. Routine (cf.Lemma 1.8.).
0
REMARKS. ( i ) L e t T be a type theory and S ( T ) as i n 1.5. Using 2.4. one can e a s i l y show t h a t T i s t h e theory o f a concrete t y p e model. J u s t d e f i n e vT (oi)= I X E I S ( T ) I 1 oiex 1 S * ( 1) = Then an easy i n d u c t i o n shows t h a t V T ( 0 ) = I x E [S(T)[I x 1
(2).
and t h a t <S* (T), VT > i s a type model whose theory i s e x a c t l y T ( c f . Theorem 1.10 o f [ 201). as defined i n [ 61). Given a s e t A, l e t XA be t h e ( i i ) (Connections w i t h , c l o s u r e o f A " h ) ( w h e r e o s A ) under ' ' + I ' and "n."Then, i f SA =<XA, 5, A , +,w >, i t can be e a s i l y proved t h a t < I S A ~ : ,E>~ ( 3 ) .
Define v : XA +DA by v (W)' P v (a)= {a} f o r a l l aeA v (b +c)= { v ( b ) + d I d c V ( C ) 1 v (br, c ) = V(b)uv(C) and v* : I S A ~ DA by +
v*(x)="
cc x
v(c).
A r o u t i n e c a l c u l a t i o n shows t h a t v* i s an embedding. As usual, i f i s an a p p l i c a t i v e s t r u c t u r e t h e s e t o f representable f u n c t i o n s over i s given by: .(DiD)= { f : D'+D 13x'D VyeD xy = f ( y ) } . Clearly, i f < D 1  , L > i s a continuous a p p l i c a t i v e s t r u c t u r e , then (D+D) C C(D,D), t h e s e t o f continuous functions from D t o D. I f we d e f i n e F ( x ) ( y ) = x  y then F i s a continuous map o f D i n t o C(D,D) (onto (D+D)). Notice t h a t ( D + D ) i s a complete l a t t i c e by t h e c o n t i n u i t y o f F.
2.5.
D e f i n i t i o n . (i) A r e  r e f l e x i v e domain i s a t r i p l e D i s a complete*lattice (2) FE C(D,C(D ,O)) and GcC((D+D), 0 ) (where (D+D)=F(D)) ( 3 ) F O G = i d (4).
(1)
such t h a t
M.COPPO ET AL.
246
!
i s algebraic i f f D i s algebraic.
domain 
REMARK. I f i s a p r e  r e f l e x i v e domain t h e n G O F i s a r e t r a c t whose 2.6. range i s i s o m o r p h i c t o ( D + D ) . I f i s a d d i t t i v e ( c o a d d i t i v e ) t h e n G O F i s a closure (projection). < I S I , * > can be t u r n e d ( i n more t h a n one way i n g e n e r a l ) i n t o a p r e  r e f l e x i v e domain. However, i t i s u s e f u l t o c o n s i d e r a p a r t i c u l a r c h o i c e o f G. B u t we f i r s t need a lemma.
2.7. (1) (2) (3)
LEMMA. L e t S be an EATS and X E I S I . Then t h e f o l l o w i n g a r e e q u i v a l e n t : a+bex bex  t a a+ b e t { c 4 I dex. t c }.
Proof. (1) * ( 2 ) . By d e f i n i t i o n o f ' I  ' ' . c + bex * a+ bex ( s i n c e c + b j a + b ) . (2) * ( 1 ) . box  ? a 3ccta ( 3 ) * ( 1 ) . By assumption f o r some I ( 5 ) A c i + d i i a + b =)
a n d V i € 1 di (2)
=.
EX
.fci
. Thus,
I by ( 2 ) * ( l ) ,V i
E
I
ci +diex
and t h e n a+ bex.
(3). T r i v i a l . 0
The lemma suggests how t o o b t a i n , g i v e n an EATS, a c a n o n i c a l G. 2.8.
THEOREM. L e t S be an EATS. D e f i n e
feC(ISI
, Isl),
G,(f)=tIa+b
L e t Go be t h e r e s t r i c t i o n o f ,G
I
".'I
bef(ta)}
( a n d F ) as above a n d s e t , f o r
. Then
FoG,:id.
t o (IS1 +IS1 ) , i s a c o a d d i t i v e
p r e  r e f l e x i v e domain. Proof. L e t s = < X , < , A , + , W > and f e C ( I S I , I S I ) . Since { t a l a e X } i s t h e s e t o f f i n i t e elements oPIS1 by 1 . 8 ( i i i ) one has f ( x ) = u { f ( t a ) ] . Thus, f o r a l l x e IS1 aex f ( x ) ={bl 3aex bef(fa)} c{bl 3aax a+be G (f)) G*(f).x. That i s fLF,G,
(f).
Note t h a t i f f e ( I S 1 + I S I ) , f = F ( z ) say, then, by Lemma 2.7, b e f ( t a ) = z * t a . Thus, i n t h i s case, one a c t u a l l y has f = FaG,,(f). Moreover, t a k e bi
E X
?ai
CE
, ieI.
Eo
F ( x ) = ?{a+ b
By Lemma 2.7
I bexfal.
Vie1 ai+bi
I t i s a r o u t i n e c a l c u l a t i o n t o show t h a t the Scott topology. n
Then, f o r some 1 , A a . +b I
E X
&
a + b e G o ( f ) =.
which implies
CEX.
i
Thus
< c with
i$0
FLid.
and F a r e c o n t i n u o u s w i t h r e s p e c t t o
I f S i s such t h a t , f o r some G ' , i s a f i l t e r Amodel i f f i t i s a &model. A f i l t e r Xmodel i s n o t n e c e s s a r i l y a r e f l e x i v e domain, c f . 4 . 1 1 ( i i ) . EXAMPLE. L e t F= be as i n 1 . 2 ( i ) , t h e n i s a c o a d d i t i v e r e f l e x i v e domain (see t h e remark a f t e r Lemma 2.13). I f we d e f i n e G'(f)= G o ( f ) u A t where f e ( l FI +I FI ) and A t i s d e f i n e d i n 1 . 2 ( i ) we can e a s i l y p r o v e
2.9.
t h a t
i s an a d d i t t i v e r e f l e x i v e domain ( j u s t mimic [ 9 ] f o r a p r o o f . ) .
247
Extended Type Structures and Filter Lambda Models
2.10.
REMARKS.
( i ) Theorem 2.8 a c t u a l l y proves t h a t
i s a continuous
r e p r e s e n t a t i o n between C ( ISl,ISI) and IS1 a c c o r d i n g t o t h e d e f i n i t i o n o f 1151, i . e . F o G& i d and 0 F 5 id.
(ii)F o & i s a c l o s u r e o f C(ISI ,IS1 ) whose range i s (IS1 +IS1 ) (use f L F o C, ( f ) f o r feC(IS1 ,I 9 ) and f = F 0 G, ( f ) f o r f e (IS1 + IS I)). ( i i i ) L e t Dk be as i n 2 . 1 1 ( i ) .
I t i s easy t o show t h a t i f is
i( Dk )s( D +Dlk. (G^(
(coaddi t t i v e ) a1 g e b r a i c p r e  r e f 1 e x i ve domain t h e n
i0((
an a d d i t i v e
(D
+
D l k )GDk 1.
Therefore I SI+lSI ) ) c Is f o r a l l EATS S. Moreover t a k e F as i n 2.9, t h e n k f( I Flk)cC( IF1 , IF1 ) k y s i n c e i s an a d d i t t i v e r e f l e x i v e domain. I t i s
k
easy t o see t h a t t h i s i s n o t t r u e f o r a l l EATS S . There a r e some s i m p l e c o n d i t i o n s on EATS which correspond t o t h e d e f i n a b i l i t y o f c l a s s e s o f c o n t i n u o u s f u n c t i o n s (among them, t h e c l a s s o f a l l continuous functions). 2.11. D e f i n i t i o n . (i) L e t D be an a l g e b r a i c complete l a t t i c e . D e f i n e Dk'ICeDI c i s f i n i t e I .
(ii)L e t D and D ' be a l g e b r a i c complete l a t t i c e s . A s t e p f u n c t i o n f a b : D + D ' d e f i n e d by
1
fab(c)'
is
b i f a& I' o t h e r w i s e
where aeDk, I X D l k and
1'
i s t h e l e a s t element o f D ' .
The f i n i t e elements o f C ( D , D ' ) a r e e x a c t l y g i v e n by t h e f u n c t i o n s Uf,.,, where a.eD b. E D ' i e I . Note t h a t faibrc)=Y {bil ai cl. I ii 1 k' k' Thus (*) iff J = { i / a i r c l # m a n d dL Ubi. J 2.12. D e f i n i t i o n . L e t S an EATS. We d e f i n e t h e f o l l o w i n g c o n d i t i o n s on S : C1) ai +bi z c + d h bi 'd
y
fcdbyfaibi
+
C2) C3)
C
I
a +b_u td and dl;w=r c a and b y A ai+bi 5 c+d=dl;w*J={i Iczai
I
l#S@3
biId.
C o n d i t i o n (*) i s c l e a r l y e q u i v a l e n t t o C3, where we t a k e ISlk and ,G
as d e f i n e d
i n 2.8. A d i f f e r e n t f o r m u l a t i o n o f C3 w h i c h w i l l be used i n many p r o o f s i s : a 1. + b i z c +d afidl;w93J#@ C I c z f ai and A bi(d. J C l e a r l y C3*C2*Cl.
9
A t y p e t h e o r y T s a t i s f i e s Cl(C2 o r C3) i f f S ( T ) s a t i s f i e s Cl(C2 o r C3). 2.13.THEOREM.Let S be an EATS. Then ( i ) satisfies C I * ( 1.~1) contains a l l constant functions. ( i i ) S s a t i s f i e s c2 9 ( I S [ + ISI) c o n t a i n s a l l s t e p f u n c t i o n s . ( i i i ) S s a t i s f i e s C3 0 (1Slt I S l ) = C ( ISI, 1st) ( i . e . < IS1 ,F,Go> i s a r e f l e x i v e domain and, t h u s , a f i l t e r Amodel). P r o o f . We p r o v e o n l y ( i i i ) . The p r o o f s o f (i) and (ii) a r e s i m i l a r and e a s i e r . Let ai+bi(c+d. Take f e C ( I S I , 6 I) d e f i n e d by f ( x ) = $ K t b i I t a i s x l = L e t now t { b i l a + x , i e I } . I t i s t h e n easy t o show t h a t G o ( f ) = t / t a i + b i .
+
1
J = { i Ic Lai
c +de Go(f).
(*).
1 ( t h u s C 5 9 a i ).
Then
L a s t l y , d7.w i m p l i e s J # a.
Go(f)* tc= t A b J i
and
3 9Id,
since
I t i s enough t o p r o v e t h a t a l l sups o f f i n i t e s e t s o f s t e p f u n c t i o n s a r e
M. COPPO ET AL.
248
r e p r e s e n t a b l e . Then t h e p r o p e r t y f o l l o w s f r o m t h e f a c t t h a t ( IS I + I S I ) i s a complete l a t t i c e . ) . We p r o v e t h a t L e t f be d e f i n e d as above (observe t h a t f = & I taifbi x f = k ( f ) = f f a i * bi r e p r e s e n t s f, i . e . t h a t 'dye IS1 xf * y = f $ bi, where
J =Iila.ey,
In
'ieI}.
1
fact
dexf'y
* 3cey
;aTbi:c+d.
Now,
if
J ' = {iI c i a . 1,' we have t h a t J'c_ J , s i n c e c L a i by C3. T h e r e f o r e
aie y. Thus J ' P O and biid d. Moreover i t i s easy t o p r o v e t h a t A b. < x * y and t h e J I f
3 bi
result follows.0
C o n d i t i o n C3 i s t h e c o n d i t i o n o f Lemma 2 . 4 ( i i ) o f [2]. Thus t h e r e p r e s e n t a b l e f u n c t i o n s o v e r < I F I , . > a r e e x a c t l y t h e c o n t i n u o u s ones. We can now g i v e some examples o f s t r u c t u r e s which s a t i s f y o n l y C 1 ( o r C2). B u t we f i r s t need a lemma. 2.14. LEMMA. L e t S be an ETS. I f t h e r e e x i s t A,BieP t h a t ASyB and Vie I A 5Z B i, t h e n S does n o t s a t i s f y C3.
( i e I ) such
(y
Bi does n o t
Proof. Observe t h a t , g i v e n any Ce p, n e e d t o belong t o P ) . n
Bi+
C=("I B i) + C G A + C
2.15. EXAMPLES. ( i ) L e t be an a p p l i c a t i v e s t r u c t u r e such t h a t Vd,eeD d  e = e. Then any (non t r i v i a l ) ETS S o v e r does n o t s a t i s f y C1. One a c t u a l l y has t h a t VA,BcD: A + A = B + B = D . (ii)L e t be such t h a t de=d. Then any (non t r i v i a l ) ETS S o v e r does n o t s a t i s f y C2, s i n c e vA,BED A+ B = D +B. C l e a r l y S s a t i s f i e s C1. ( i i i ) L a s t l y , we show an ETS which s a t i s f i e s C2 b u t n o t C3. L e t < x ,  >be t h e Kleene a p p l i c a t i v e s t r u c t u r e d e f i n e d by nm = { n 1 (m)
s a t i s f i e s C2. L e t A,B,C,E
. Actually
be non empty subsets o f 2 and EP 2. Then,
i f A + B G C + E , c l e a r l y BGE. Moreover l e t p e g \ E and q E " I k l ( x ) = i f x= r t h e n p e l s e q " i s such t h a t ke A+B b u t i s g i v e n by t h e C2 i s s a t i c f i e d . A X p l e T S o v e r < & , a > 1.4. Namely, by t h e I 1 Recursion Theorem t a k e no such t h a t
m e 2 . Then
{ A G yl
any ETS o v e r
B. I f r e C \ A, t h e n k t C+E. Thus G A and same argument used i n { n o X m ) = no, f o r a l l
no E A } i s an ETS and does n o t s a t i s f y C3 by Lemma 2.14.
A l s o t h e e x t e n s i o n a l i t y p r o p e r t y o f qISI,.> has an easy c h a r a c t e r i z a t i o n i n terms o f t h e p r o p e r t i e s o f S As u s u a l , an a p p l i c a t i v e s t r u c t u r e i s e x t e n s i o n a l i f f VG D a. c = b  c * a=b f o r a,beD.
.
2.16. THEOREM. L e t S*X, 2, A , +,u> be an EATS. ( i ) V'ze I S l x . z = y  z 0 (Va,beX a + b e x * a + b E y ) , f o r x , y l S I . ( i i ) . ISl,*>is e x t e n s i o n a l i f f VaeX 31 a * A b + ci.
I
Proof. ( i )
()
a+bex
i
b e x  f a (by Lemma 2.7) * be . f a * a+iey. Immediate f r o m t k e d e f i n i t i o n o f ''.''. ( * ) Let x a = f { b + c l a is
9 bi+
( c )Easy
3.
extensional.
Thus
ae xa,
that
is
By ( i ) x a = f a ,
3 1 Vi € 1 bi+ ci e xa
c. 5 a. 1
from ( i ) . 0
EMBEDDINGS AND ISOMORPHISMS
D e f i n i t i o n . L e t D be an a l g e b r a i c complete l a t t i c e . D e f i n e (i) 'c= {x I c L x } f o r c EDk, t h e cone o v e r a ( f i n i t e ) element.
3.1.
2 49
Extended Type Structures and Filter Lambda Models (ii)
K ( D ) = { z l ceD2.
( i i i ) C ( D ) as t h e c l o s u r e o f K ( D ) under f i n i t e union. As w e l l known, K(D) i s a b a s i s f o r t h e S c o t t t o p o l o g y on D. 3.2. REMARK. C(D) i s i f f C ( D ) i s c l o s e d under
c l o s e d under 'In''., r e f l e x i v e domain.
(1) (11)
Proof. de
+
( i ) We p r o v e t h a t Va,trDK
*a +b*=)F(d):
".
.
i s a coaddittive "
I
a + b = G(fab). O b v i o u s l y G ( f a d s a + b .
fab
*G(F(d)) !G(fab) (by c o a d d i t t i v i t y ) . *dlG(f
,d
(ii)D e f i n e a r b by : a L b = >+E, Notice t h a t b d e f i n i t i o n (1) f,bfF(d) * deZ+"b * arbgd
where a , b D
K'
and G ' ( f )  C H a r b l f a , C f ) .
Moreover
M.COPPO ET AL.
250 and
f abC f * a c b LG'(f) (2) Thus F(G' ( f ) ) c K f a'b I f a b & F(G' ( f ) ) 1 = U [ f a b l a + b L G ' ( f ) l , by ( 1 ) CKfabl fabEf} f.
by ( 2 )
=
G'(F(d))=Uabb I f
CF(d) } abU a r b I a+b E d } , b y (1) d.
3.6.
REMARK. Since any
o f Theorem 3.5 i t
i s an e x t e n s i o n a l r e f l e x i v e domain, f r o m t h e p r o o f
f o l l o w s t h a t Va,b e (D,
)k a c b = f a E
From t h e p r e v i o u s r e s u l t s we o b t a i n t h e isomorphism o f any a l g e b r a i c c o a d d i t t i v e r e f l e x i v e domain w i t h t h e f i l t e r Amodel, b u i l t on i t s compact cones. 3.7. D e f i n i t i o n . [ 151 An isomorphism between t h e r e f l e x i v e domains and = < I S I , F,G,>. K K Proof. S Ki s an ETS b y Theorem 3.5. Since < D , F ' >  < I S K I , F > (by Theorem 3 . 3 ( i i ) ) t h e range o f F i s C(IS 1,l Sd ) I i . e . < I S I,F,GO> i s a r e f l e x i v e domain. K K I n 1151 Sanchis n o t i c e s t h a t g i v e n two c o a d d i t t i v e r e f l e x i v e domains, o n l y one o f t h e c o n d i t i o n s o f 3.7 s u f f i c i e s t o have t h e isomorphism. So we a r e done, s i n c e < I SKls,F,Go> i s c o a d d i t t i v e by Theorem 2.8 and c o n d i t i o n ( 1 ) o f 3.7 h o l d s , b y Theorem 3 . 3 ( i i ) . 0 Another i n t e r e s t i n g c l a s s o f embeddings i s d e f i n e d c o n s i d e r i n g f i l t e r s o f EATS b u i l t from tvoe theories. F o l l o w i n g [ f i ' l an element U E I FI i s a c l o s u r e o e r a t i o n i f f i t s a t i s f i e s : G , ( i d ) L u = u 0 u ( where u 0 u =G,(Az. u.(u.z)T). 3.9. (i (ii)
THEOREM. L e t T be a t y p e t h e o r y . Then one has % < I S ( T ) I ,  , c _ > i s i s o m o r p h i c t o t h e range o f a c l o s u r e o p e r a t i o n u e I F I .
Proof.
.
.
( i ) Observe t h a t a b s t r a c t f i l t e r s o f
S ( T ) are abstract f i l t e r s o f
then, a r e c l o s e d u n d e r a p p l i c a t i o n . Thus < I FI , * ,g >.
F and,
i s a substructure o f
25 1
Extended Type Structures and Filter Lambda Models
( i i ) As p o i n t e d o u t i n 1.6, z T extends 2 0 . D e f i n e u = t Io+TI O < T T I E I FI a n d + A d S ( T ) I as t h e f i l t e r generated by t h e s e t o f t y p e s A ( n o t i c e t h a t u i s c l o s e d under lo w h i l e + A i n c l o s e d under
3.We p r o v e t h a t u*A=+A.
+Ac_ u . A i s t r i v i a l . F o r t h e r e v e r s e observe t h a t T E
u.A3Ue
A
*
u
s
O+T
* 30s A * 30 E A * 30 E A =. 3 0 c A
31 V i e 1 u ~ ~ ~ T o ~ ~& + 31 V i e 1 0 ~ i ~ ~ a n d 3 J uz0 c _ I3 u i
33 ui
U L ~ /Jui T
T ~ A .
,
OT+ T
~
~
~
,I~T~ , f (o r~F T satisfies
C3
zT$~iiO~
s i n c e < extends lo T
Obviously u _ > i = t { o + . r l u i o T I
and u = u

3
u. C l e a r l y S ( T ) i s t h e range o f u. 0
Theorem 3.9 proves t h a t i s " u n i v e r s a l " ( i n t h e sense o f [ 191) f o r a l l f i l t e r domains. R e c a l l i n f a c t t h a t each such domain i s t r i v i a l l y isomorphic t o an EATS g i v e n by a s u i t a b l e t h e o r y (see what p o i n t e d o u t a f t e r 1.6). We can a c t u a l l y p r o v e t h a t any a p p l i c a t i v e s t r u c t u r e can be embedded i n t o . 3.10. THEOREM. L e t % .
b e a ( c o u n t a b l e ) a p p l i c a t i v e s t r u c t u r e . Then
Proof. L e t A = { a . I i L l } and x A = { $ i i $ j + $ d a i * a j = Emb: A + I S ( T z
)I by
Emb(ai)=
t$i
a h } . Define
(iL1).
A
Emb(ai)* Emb(a.)= Emb(ai. a .). _> i s t r i v i a l and C_ i s g i v e n J J by t h e m i n i m a l i t y o f iZA, as d e r i v e d f r o m x A ( f o r t h i s some b o r i n g c a l c u l a t i o n s
We c l a i m t h a t
a r e needed. We l e a v e them as an e x e r c i s e ) . Moreover < I S(Tz ) I ; > k < l A
FI
>;
by Theorem 3 . 9 ( i ) .
0
The c o n d i t i o n on t h e c a r d i n a l i t y o f A may be dropped j u s t t a k i n g enough atoms, i . e . t a k i n g A t l a r g e enough and c o n s t r u c t i n g T f r o m i t as i n 1 . 2 ( i ) . 4.
FILTER AMODELS
As a l r e a d y p o i n t e d o u t , any EATS s a t i s f y i n g C3 y i e l d s a f i l t e r Amodel. A c t u a l l y any such Amodel i s g i v e n by a r e f l e x i v e domain, i.e. i t has t h e s t r o n g p r o p e r t y t h a t any c o n t i n u o u s f u n c t i o n i s r e p r e s e n t a b l e . T h i s i s more t h a n what i s r e q u i r e d by an a p p l i c a t i v e s t r u c t u r e t o y i e l d a Amodel. Theorem 4.8 c h a r a c t e r i z e s EATS S such t h a t < S,F,G > i s a f i l t e r Amodel. Theorem 4.11 g i v e s a f i l t e r Amodel, which i s n o t a r e f l e x i v e domain. F o r t h e n o t i o n o f (expanded) combinatory a l g e b r a and Amodel we m o s t l y r e f e r t o [ I 1 s [ 9 1 9 [I01
.
4.1. D e f i n i t i o n . L e t S= <X,(,A,+,W> be an EATS ( r e c a l l t h a t a +b+ c stands f o r a + ( b + c ) ) . D e f i n e K = t { a + b +c I CE t a 1 S = t { a + b +c+d I dcta.tc(tb.tc)} E * t { a  . b +c I CE t a  t b l .

Note t h a t
K,
S a n d 5 have been d e f i n e d j u s t u s i n g G*
o f 2.8.
M. COPPO ET AL.
252
4.2. LEMMA. L e t S be an EATS. Then x  z (y.z) ~ _ S  x . y  z and x.y@x*y. ( i ) Vx,y,ze I S I x g : x . y , i s a combinatory a l g e b r a , t h e n KSK and S C S . Moreover, f o r ( i i ) I f < lSI,,S,K> I=SKK, a+ b € 1 0 a 2 b. Proof. ( i ) By FOG*,
i d (see 2.8).
( i i ) Observe t h a t K . t a  t b = t a i m p l i e s by 2.7 a+b+ccK f o r S. Moreover a + b e I 0 b E I  t a = t a . 0

f o r a l l ceta. S i m i l a r l y

4.3. THEOREM. L e t s be an EATS. I f t h e r e a r e S, K such t h a t < I S I , ,S,K > i s an expanded combinatory a l g e b r a t h e n a l s o d s l , , $,E > i s an expanded combinatory algebra.
Proof. Immediate f r o m 4.2. THEOREM. L e t S = < X , L , A , + , w > 4.4. ( i e 1 ) o n e has(*) /\(bi+ci)+bi+ciza+b*3J
be an EATS. Assume t h a t f o r any a,b,bi,ci a 2 A d . e .&I. J J 3I Then, i f i s a combinatory a l g e b r a , < l S I , . , E > i s a 1model.
EX
P r o o f . F o l l o w i n g [ l o ] , we j u s t need t o show t h a t ( 1 )g.x.y = x  y ( 2 ) vz x.2 = y.z =$ g. x =g.y (3) E.4 = E . As f o r (17, n o t e t h a t g=S(K(S&)). Then use 4.3. As f o r ( 2 ) , observe f i r s t t h a t a + b e E 31 ? a i + b i  + c i ( a + b f i V i E I a 1. i s a xmodel, t h e n E =S(K(SKK)) ( c f . 4.1) and, by d o e s n ' t need t o 4.2,s C _ E . However, a l s o i f < l S I ,  , ~ > i s a xm<delS be a m o d e l , f o r < IS I , ,K,S_> may j u s t be a combinatory a l g e b r x ( o r a x  a l g e b r a ) . Each t y p e t h e o r y T induces a system o f t y p e assignment, i n t h e sense o f [ 2 1 , f o r t h e s e t A o f xterms. By t h i s , Theorem 4.8 c h a r a c t e r i z e s t h e t y p e t h e o r i e s which y i e l d 1models. N o t a t i o n and concepts a r e m o s t l y f r o m [ 21. I n p a r t i c u l a r i f CJ E T and M E A , then UM i s a statement, where u is t h e p r e d i c a t e and M t h e s u b j e c t . A basis i s a s e t o f statements w i t h o n l y v a r i a b l e s as s u b j e c t s . 4.5. D e f i n i t i o n . L e t 7 be a t y p e t h e o r y . The (extended) t y p e induced by T i s d e f i n e d by t h e f o l l o w i n g n a t u r a l d e d u c t i o n system
assignment
253
Extended Type Structures and Filter Lambda Models
( + ) i f x i s n o t f r e e i n assumptions on which T M depends o t h e r t h a n T
W r i t e B t
oM i f
OX.
oM i s d e r i v a b l e f r o m t h e b a s i s B i n t h i s system.
4.6. D e f i n i t i o n . L e t T be a t y p e t h e o r y and S ( T ) be t h e EATS d e f i n e d i n 1 . 5 ( i v ) . F o r any map 5 f r o m v a r i a b l e s o f A t o IS( ~ ) and 1 M E A. define: (i) B =Ioxloe s(x)}
5
(ii) (II b y i n d u c t i o n on t h e s t r u c t u r e o f M) u x f =
UPQ 1 = F ( EPII [Ax. P
T
( UQII
T
I T = ( ,G he E 1 S( T ) I .[ P 1 ) 5 c [ x/el
(see 2.8).
( T h i s i s w e l l d e f i n e d , by t h e c o n t i n u i t y o f F and Note t h a t i f i s a xmodel,
I oy
Thus V i
C_
. By i n d u c t i o n on M.
I
I
, by
induction
T
, by
rule ( < ) T
Bd x / t a . l ~ BiP 1
* BPUIaix}t~;P T
* B ta.+Bi 5 1
Xx.P
, by (
The r e s u l t f o l l o w s b y u s i n g (
II J
i s the
[MI
T
5
= Io l B
T
5
toM}.
The o n l y non t r i v i a l case i s M  X x . P .
r
E
then
E B and y z x l .
THEOREM. L e t T be a t y p e t h e o r y . Then
Proof.
).
A
+I).
I ) and ( z T )
(standard)
M. COPPO ET AL.
254
.
3
T By i n d u c t i o n on t h e d e d u c t i o n B g t  oM. We j u s t check when ( + I ) i s used.
The r e s t i s t r i v i a l . Note t h a t i f
t ax1
t h e n we have, by a s h o r t e r d e d u c t i o n , gives the result. 0
BF[x/+alg 8p. The
i n d u c t i v e hypothesis
L e t G o be as i n 2.8. THEOREM. L e t T be a t y p e t h e o r y . Then i s a xmodel * T T lB/X t U+T xx.M * B / x ~ (UXI F TM). Proof. =*. R e c a l l t h a t G o i s t h e r e s t r i c t i o n o f G, t o ( I S ( T ) I + I S( T ) I ) , t h e
4.8.
IS( T ) I t o
IS( T ) I,
i s representable.
Thus t h e
representable functions.
By assumption,
which i s d e f i n e d by a
xterm
semantics o f xterms i n
IS( T ) I i s d e f i n e d e x a c t l y as i n 4 . 6 ( i i ) ,
.
use G o i n s t e a d o f G, Let 4.7
g ( x ) = f { u J u z ~o r B
r
B/x+u+T
T
oxeB}. O b v i o u s l y B F p N * B I\x.M]
XX.MU+TE
any f u n c t i o n f r o m
(using constants),
T
I~N gB
Te
F(Go(f))(fu) f o r T
I
f=AeeIS
(T)
T
* B
/X U { U X } t T M . gB =. The p r o o f o f Theorem 3.5 i n [ 2 ] remains v a l i d , c which r e q u i r e s t h e g i v e n c o n d i t i o n .
4.9. Oefine
except f o r
point
(iii),
we can now g i v e a c l a s s o f f i l t e r M o d e l s , which a r e n o t
D e f i n i t i o n . ( i ) Choose 0
c*=
b y Theorem
r
q x / t u 1‘:’?ince
Using Theorem 4.8, r e f l e x i v e domains.
. Then,
where one may
{ ULU[ $/p]
IUE
A t and PcT such t h a t
T}.
( i i ) I * a n d ? i s short f o r (iii)u ~ i Tf f e i t h e r (1) u 3 T
zz* and &(E*),respectively.
0 does n o t o c c u r i n
p
.
Extended Type Structures and Filter Lambda Models 4.10. (ii)
ui*
LEMMA. ( i )
T
=)
o ~ A a + 6< * A y
. ,...,
d O / p I ~ * T[ $/PI + 6 = ~ * 3pl
pn
255
LT.
o b p l z . . . < p
I i i J i in and each p (1 I h I n ) i s an i n t e r s e c t i o n o f arrows. Fi B&M~=) BL +/PI P T [ 4/01 M . ( i v ) vie1 B/XU{a.X} 6 f3.M and ai+Bi & A y . 4 . * VjeJ B / x U { y . x } p 6 . M . 1 1 J J J J J