Souslin Problem

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

405 Keith J. Devlin H,~vard Johnsbraten

The Souslin Problem

Springer-Verlag Berlin-Heidelberg • New York 1974

Dr. Keith Devlin Seminar fer Logik und Grundlagenforschung der Philosophischen Fakult~t 53 Bonn BeringstraBe 6 BRD Dr. Havard Johnsbr&ten Matematisk Institutt Universitetet I Oslo, Blindern Oslo 3/Norge

Library of Congress Cataloging in Publication Data

Devlin, Keith J The Souslin problem. (Lecture notes in mathematics, 405) Bibliography: p. 1. Set theory. I. Johnsbraten, Havard, 1945joint author. II. Title. III. Series: Lecture notes in mathematics (Berlin) 405. QA3.L28 no. 405 [QA248] 510'.8s [5!i'.3] 74-17386

AMS Subject Classifications (1970): 02-02, 02 K05, 02 K25, 04-02, 04A30 ISBN 3-540-06860-0 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-38?-06860-0 Springer-Verlag New York • Heidelberg • Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1974. Printed in Germany. Offsetdruck. Julius Beltz, Hemsbach/Bergstr.

Acknowledgements

The last five chapters of this book are based almost exclusively on a set of notes written by Devlin during the S11mmer of 1972.

During this time, he was living with,

and almost en-

tirely supported by his wife's parents, Mr and Mrs H. Carey of Sidcup, Kent, to whom thanks

should now be put on record.

The manuscript was typed by Mrs. R. Msller of the University of Oslo, for whose care and patience we owe a considerable debt.

CONTENTS

Chapter I. II.

Preliminaries

I

Souslin's Hypothesis

8

The Combinatorial Property

23

Homogeneous Souslin Trees and Lines

32

Rigid Souslin Trees and Lines

46

Martin's Axiom and the Consistency of SH

5~

VII.

Towards

68

V~I.

Iterated Forcing Jensen Style

74

How J e n s e n K i l l e d

97

III. IV. V. VI.

IX. X.

ConZFC

Con(ZFC + C H + SH) : A False Start

~

a Souslin Tree

Con(ZFC+CH+SH)

113

APPENDIX. Iterated Souslin Forcing and ~ ( ~ )

122

REFERENCES

426

GLOSSARY OF NOTATION

128

INDEX

CORRECTION

A n Israel£ mathematician

called Uri Avraham has pointed

t~at~ in Chapter

IX, the claim straddling

false as stated.

To be precise,

define,

for C a certain

order-embedding

~a+2

(although

on page

should have

In fact,

given an Aronszajn

closed unbounded

subset

tree ~, we

of ~ ,

a ('generic)

to all of ~° As we have set things up this

a slight modification

to the definition

105 would in fact make our claim said is not that our particular

all of @, but

105 and 106 is

of TIC into Q, and claim that this order-embed-

ding can be extended is false

pages

out

that it easily

this is perhaps

in terms of our original

true).

of

What we

embedding

extends

to

gives rise to such an embedding.

more easily definition

seen

(via Theorem II.5)

of special

Aronszajn

(Page

15)= By means of our embedding Um< Am, where

of TIC into Q, we may write TIC =

each A m is an (uncountable)

~:aml~ be a or~e-one enumeration ~ be the monotone e a c h ~ x e T c , let s x = able,

let

lysTlc

1- , , ( W z ~c~< & )~'-u { U - -, m - , m

Keith J. Devlin 1 4.9 • 74

of A m , each m, and let <m~l

+ I I x ~ ( z , x

The c_onstructible hierarchy,

If

2

so is

the a s s u m p t i o n

ZFC + V

= L .

(Hence,

any result w h i c h

V = L

will be c o n s i s t e n t

relative

to

.)

Theorem Let

2

(Condensation

lim(m) .

Theorem Let

If

X~

Lemma)

Lm

then

X ~ L~

for a unique

6 ~ m .

3

a I = w~

If

X~La2

If

A

, a2 = w~ .

, then

is a g i v e n

sets a n a l o g o u s l y allowing

A

XNLal

If

X~Lal

= LB

for some

set we can define to the d e f i n i t i o n

to figure

of the d e f i n i t i o n s .

, then

La+I[A]

for some

(LaEA] I a E 0 n )

of the c o n s t r u c t i b l e

will

6 ~ at°

B ~ ~I

a hierarchy

in the c o n s t r u c t i o n

(Thus

X = L~

as a u n a r y consist

of

hierarchy,

predicate

of all subsets

by

in all of

-3-

L [A]

which

structure

are f i r s t - o r d e r

(LaCA], E , A N L a [ A ] )

are g i v e n in [DeI], rems 2 and 3

where

.

from m e m b e r s

Full d e t a i l s

proofs

of

L [A]

of this

of the f o l l o w i n g

in the

construction

analogues

of theo-

may be found.

Theorem

4

Suppose

A ~ w L [ A ] . If

nals

definable

X-~L

[A]

, then

X T L ~ [ A 0 y]

ordi-

for unique

¥ < 8 _< ~ •

Theorem Let

5

ml = "I

for some

' ~2 = w

8 ~ ml

If

, A ~ ~I X

(2)

disjoint

S

x E S

from

D

has cardinality

versely,

x 6 S

2w .

there

x

from

S

defined by

fn(X)

= xn .

(Since

S

is complete,

subsets of

For

in

sequence

ISI ~ 2 w . ConS

into

n E w , let

Then the functions

by a theorem of J. Ball

~w

(D) ~

fn

2w.)

sequence fn : S ~ S be

are not all con-

[Ba I] .

question w h i c h h i s t o r i c a l l y

Mary Ellen Rudin

[Rul]

has shown that

exists a Souslin line, then there exists a normal Hausdorff

space w h i c h is not countably paracompact. C.H. Dowker

[Do I]

pact and normal

iff

we obtain that in (4)

sequence

D , a set of cardinality

In conclusion we m e n t i o n one topological

if there

of cardi-

as the least u p p e r bound.

with lub x .

SH .

In

Consequently:

exists a strictly i n c r e a s i n g

<x i l i e w>

has been connected with

D

there exists a strictly increasing with

S ,

is an uncontable

by (I), there exists an injection from

For each

tinuous,

[<x~,xv+1> I v < w l ]

of

conjecture.

has a dense subset

open intervals).

the set of all countable

(3)

S

a natural

is a strictly increasing

(for otherwise

family of pairwise

(7)

(x v Iv < ~>

If now

exists a Souslin line if and

There

that a topological X x I ZFC + xSH

is normal

space (I

space

X

is countably p a r a c o m -

is the real unit interval),

the f o l l o w i n g

exists a normal Hausdorff

not normal.

Together with the result of

is provable: X

such that

X x I

is

-11

Recently, ~SH

M.E. R u d i n [Ru 3 ]

, i.e. as a t h e o r e m in

-

has f o u n d a proof of (4) without a s s u m i n g ZFC

alone.

2. Souslin trees A tree is a poset [yET

Iy~x]

~ = (T,~)

is well o r d e r e d b y

the height of

x

we m a y define

ht(x)

For

a E On

We write For

such that for every

, ht(x)

, the

Tla

x E T

~

.

.

be a tree. ~

b y i n d u c t i o n as follows: ~

Tx = [yET1 ht(T)

x ~y]

T .

under

ht(x)

is the set

for the r e s t r i c t i o n of

sup[ht(x) I x E X ]

T

, is the o r d e r type of

~'th level of

we set

Let

x E T , the set

T

x E T ,

~

(Hence

= [xET

lht(x) =~].

Tla = U ~ a T ~ -

X ~ T

is called the length of

For

= s u p e r ( y ) lynx].)

to the set When

x =

we let ~ .

ht(X)

=

(From now on,

w h e n there is no danger of confusion, we denote a tree simply b y

T =

~T,~) • )

Two elements

x,y

of a tree

t h e y are incomparable, subset of x E b

T .

and

T

written

A branch

b

y ~ x , then

are c o m p a r a b l e if x ~ y xly

.

.

If

y~x

; if not,

A chain is a l i n e a r l y o r d e r e d

is a chain w h i c h is

y E b

or

ht(b)

~-closed,

= a , then

b

i.e. if is an

a-

branch.

An a n t i c h a i n is a set of p a i r w i s e i n c o m p a r a b l e elements of

A branch

(resp. an antichain)

in a l a r g e r b r a n c h

A tree T

T

ITI = w I

Ta

(resp. antichain).

Ta

ITI = ~I

is countable . T

T

is A r o n s z a j n or Souslin,

is an antichain~

and every chain in

is called a S o u s l i n tree if

and every chain and a n t i c h a i n in

N o t e that if each

is called maximal if it is not i n c l u d e d

is called an A r o n s z a ~ n tree if

and every level

T .

T

then

is countable.

ht(T)

= wI .

every S o u s l i n tree is Aronszajn.

Since Aronszajn

trees can (in the p r e s e n c e of the axiom of choice) be p r o v e d to exist (c.f.

[ J e l ], [ J e ~

, or [Ru2]). It is often convenient that the trees

satisfy c e r t a i n "normality" properties.

So let us call a tree

T

a

-12-

normal tree of length (i)

(ii) (iii)

ht(T)

T

= ~

has a unique least point,

each non-maximal

and

(v)

If

T

point

if:

(i.e. points

called the root of

T .

has at least two immediate

y

such that

ht(y)

= ht(x) + I

x ~ y ) ,

each

at each greater level

8 -branch has at most one immediate

< m ,

successor w h e n

is a limit ordinal,

each level

is normal,

Tv

is countable.

then

will be uncountable).

ht(T) ~ ml

chains.

without u n c o u n t a b l e

antichains.

Souslin tree.

(otherwise,

by (iii) and (iv), T I

A normal A r o n s z a ~ n tree is a normal mq-tree

without u n c o u n t a b l e

antichain,

m-tree)

x

each point has successors

8 (vi)

(or a normal

,

successors

(iv)

~

A normal Souslin tree is a normal

w I -tree

Then a normal Souslin tree is indeed a

(For an u n c o u n t a b l e

chain gives rise to an u n c o u n t a b l e

again by (iii).)

Theorem 3 Every S o u s l i n tree

T

Souslin tree

Similarly for A r o n s z a j n trees.

Proof:

T*

.

Remove f r o m

can be normalized,

T

the points

x

i.e.

for which

T

contains a normal

Tx

is countable,

then remove those points with only one immediate The resulting tree, T' , is still of length By induction,

pick out one point

in

To

sulting subtree,

struction applies also for A r o n s z a j n trees.

due to E.W. M i l l e r

ml ' hence Souslin.

in

T" 9 is a normal Souslin tree.

The long list of definitions [Ni I] :

successor.

and one point

every b r a n c h of limit length with extensions

and

T'

extending The r e -

The same con-

|

above is justified by the f o l l o w i n g fact,

-13-

Theorem 4 Souslin's h y p o t h e s i s tree,

to the n o n - e x i s t e n c e

of a Souslin

i.e. there exists a Souslin line if and only if there exists a

Souslin tree. cardinality

Proof:

is equivalent

Furthermore,

of

wI

Suppose S

every Souslin line has a dense subset

S

is a Souslin line.

is countable,

it is not dense in

union of a n o n - e m p t y open intervals.

collection

D

in

By induction, I(U~< a D ~ ) in

IS

assume

and let

Indeed,

T

I(D)

D

o

is the

of pairwise

disjoint

is countable,

since

is defined for consist

We claim that

S

~ < ~ .

Set

S.

I~ =

of one point from each interval and p a r t i a l l y order wI

T

b x re-

is a Souslin tree. with

"normality proper-

hence it is enough to show that every anti-

is countable.

f a m i l y of pairwise

S - D

of

consist of one point from

is a tree of cardinality

T

whence

D

above denotes the topological

T = U <Wl Im

(iii) satisfied,

chain in

Let

D~ Da

Now let

verse inclusion.

ty"

S. )

(~

S,

I(D)

And of course,

has the Souslin property. closure of

Observe that if a subset

But an antichain in

disjoint

open intervals

in

T

is just a

S , hence

count-

able. N o w we look at that

D

D = U~<wl D~ .

is dense in

< w1

S .

If not, let

there is an interval

Ja o J~+1 chain in

for each

a , so

T , impossible:

Clearly,

Jm E I m

tree.

.

containing

For each p .

But

is an u n c o u n t a b l e

Hence we have shown that a Souslin

suppose that



In fact, we may also assume that

lowing strengthening

p E S-D

[J~a

Then let

Let

such that

Tm



such that An'

b = it [~n ( t ~

that

ht(sn) > ~ n ' Sn)]. )

consist of the sequences

Ub s

for

s £TI~.

is defined.

antichain

of

T .

be a countable, A E M .

cally satisfied meters.

Define

So -->T s ~ and

T

Now

wq , so it remains

is countable. T,A c L

, so

Let

elementary

We also want

since b o t h are definable

Now we use Theorem 1.2.

be a w2

submodel of

T,w I E M ,

A

to

T,A E L

wI M

be a

such b r a n c h in the canonical w e l l - o r d e r -

show that every antichain of

Let

A n , n E w,

SUPnEw ~n = a "

is obviously a normal tree of length

maximal

antichains

Let

lies above a point in

be the least

s

6~ < w~ , so

U T8. Hence, for every s E Tim there exists an 8<m b going through s and m e e t i n g every A n (Such

a b r a n c h m a y be obtained as follows.

for each

Then

(L 2 , E )

such

but this is automatiin

L

w2 It follows that

w i t h o u t parawI O M

is

-q8 -

transitive,

so let

e = wI A M < w I

, the collapsing isomorphism,

(by

x E M .

So

Hence if

induction).

Lwq

ON, wI w(wq) = m .

Thus

= Ty

y

for

~(A)

c

Now,

y < wI .

,

But

e

= ~"(AON)

= AAN

of

Tie

and in

= ANT}a

.

But since

is maximal also in

Note:

By modifying

the

some

A ~ w I , then

~SH

If

A final remark.

e = ~(Wl)

ZF

L6e.

antichain of

T e , every point in

Corollary:

M,

is countable in

is a maximal

~(x) = x

is definable Ty E N

in

for y < ~.

,

N o w we are ready to conclude the proof. Lw2

so

Ty'S

.

Since

AcT,

so

~(A)

able in

T¥ Thus

is defined as the u n i o n of the

TI~

x c M,

= w " ( x N M)

= ~(Uy~.4~l Ty)

= Uy

~

TIC ~ t h e n there is a u n i q u e i s o m o r p h i s m

<S,~>

such that

~ " I s = Io(s)

for all

s E TIC

.

Lemma 8 Let

~ :<S,~)

~

<S,~>

such that if

s E Ta

Is , .

~

set

Hence

~(s)

s'

assume

: T~C ~ on

and

s' E Tm

a u t o m o r p h i s m on

and let

a : <S,~)

there exists a closed u n b o u n d e d TIC , and

TIC

.

~"I s =

TIC

if we

~

~

<S,~)

C ~ wI

.

such that

reverses the l e x i c o g r a p h i c a l o r d e r i n g

On the o t h e r hand, b y the p r o o f of T h e o r e m 3,

a < ml

Thus let

s,s'

such that if

E TIA

a ~ v < ht(s) , then

such that

~(s) ht(s'),

is h o m o g e n e o u s .

Indeed,

For

let

~x

be the l a r g -

= s~8

or

x~8

x 6 T

s u c h t h at

x~

let

= s'~B

.

by

{ i s'

if ~ v < x~x if v
~

we h a v e

= a

and

lim~

,

such that Or

s

= I -

S t

for

% > ~ .

H e n c e we o b t a i n (3)

If

a ~ wq

is c l o s e d u n b o u n d e d

automorphism s u c h that or Thus

T

If we let

mq

is c l o s e d u n d e r

ble.

|

T~A , then there i) h t (s)

> v ~ a ~ s

o

is an

is an

a < mq

> v ~ m - s v = o(s v) = 1-o(s

)

automorphisms.

S = S T , it f o l l o w s

Souslin line with S

either

ii) h t ( s )

has

on

and

exactly

wj

the m a p p i n g

that

S

is a h o m o g e n e o u s

automorphisms. s'-~s

.

Hence

But in this case, S

is r e v e r s i -

Chapter V

RIGID SOUSLIN TREES AND LINES

1. A rigid Souslin tree A poset ism on

X = <X, ~ > X .

is called rigid if

We now prove that in

idIX

is the only automorph-

L , rigid Souslin trees exist.

Theorem 1 Assume ~

Proof:

.

Then there exists a rigid Souslin tree.

Let

A ~ wI

IV.I.

and

Ma

(m < w I)

We define a standard tree

T~+ 1 = [ s ~ ( i ) with

Is E T

lim(~)

To define

T

^ i E 2]

as follows.

for

a < wq

As an induction hypothesis,

T~

we force over

=

E M a,

M

A p:a-

G c]P

be the least

rify.)

(in L[A])

is countable in

thesis that

TI8 E N B

Define for

Now let

assume

N -generic

Ma+ I , G E Na+ 1 .

for all

lim(B)

set for

ii) iiD iv)

bn

b n ~ bm each

bn

for is

a-branch

of

Tla,

n ~ m, N -generic for

if

nq,...,n m

are distinct,

bn I

x bn 2 × ... Xbnm

is

~

.

will be trivial to ve-

n E

is an

.

(Hence the hypo-

Claim: Each

TIa E M a .

(pn~qn)

bn = [Pn I p E G ]

i)

a < w

TI~)],

~-~ dom(p) ~ dom(q) ^ Vn E dom(q)

Na

= {d} and

O

with the set of conditions

p ~q

Since

.

T

defined as follows:

: [plSa c ®(lal a ,

since

and

d E Is,

Hence we have p r o v e d that

is rigid.

If now

S

was reversible,

then, b y Lemma IV.8 , there w o u l d

have b e e n an antilexicographical

automorphism

some closed u n b o u n d e d

But then

is impossible,

since

C ~ wI id

~

on

TIC

for

~ = ida(TIC) , which

is not at all antilexicographical

I

One might be t e m p t e d to conjecture not be reversible.

that a rigid ordered continuum can-

This is not so, however.

We now construct f r o m ~

a rigid Souslin line w h i c h is reversible. Theorem 3 Assume

~

.

tomorphisms,

Proof:

Then there exists a Souslin tree n a m e l y the identity on

T

T

= dom(s)

and

with exactly two au-

and the m a p p i n g

s E 2 <wl , ~

We recall at once that for dom(s)

T

s~ = q - s v

for

s~'~s

.

is defined by:

v < dom(s)

The tree

is d e f i n e d as the rigid tree, but with the f o l l o w i n g modifi-

cation:

If

we now define every branch

lim(~) , then T

to be

{s I s E bn}

Souslin b y Lemma IV.2. that

T

~ , G , b n (n E ~) [Ub n I n E w ] is

are as before.

U [~n

Ma-generic

for

InEw] T

But

" Of course,

, so

T

is

It is also clear from the c o n s t r u c t i o n

is closed under the a u t o m o r p h i s m

sr~s

.

As in the final part of the proof of Theorem I, the assumption that

~

is different b o t h from the identity and the m a p p i n g

-

s~s

-

readily leads to a contradiction.

as before. ther

~(x)

Then for every = x

x o I o(x o) ~ Xo that

o(x) ~ x ,

or "

By m o d i f y i n g the proof, r e p l a c e d by

TIC

o(x)

For let

x E T m , a(x)

= ~ .

Then,

hence

contradiction :

T

50

Let

taking o(x)

be

E Mm[x] , hence ei-

x ° E Tim

x E T m,

= x .

m = mm

be such that

x • x o , we have

But then

°(Xo)

= Xo ' a

|

one finds that this theorem is also true with

whenever

C ~ wI

is closed unbounded.

Theorem 4 Assume ~ .

Proof:

Then there exists a Souslin line (i)

S

is rigid,

(ii)

S

is reversible.

Let

T

be the tree of Theorem 5.

note above and Lemma IV.6

Set

we get that

as in the proof of Theorem 2.

to

such that

S = ST . S

<S, ~ > .

From the

is rigid,

Furthermore,

obviously gives an i s o m o r p h i s m from which is isomorphic

S

(S, ~ ) ( S

exactly

the m a p p i n g

br~b

onto a continuum

is obtained by remo!

ving the right-hand

sides of all gaps from

Chapter IV, section 2.)

ST

as defined in

|

3. S o m e final remarks We know from deed,

if

T

~

that there are two n o n - i s o m o r p h i c

Souslin trees.

is the rigid Souslin tree from section 1, then

In fact, by an easy argument, that there are

2 wl

Jech

non-isomorphic

LJe 3]

has p r o v e d that

Souslin trees.

to the '~classical" result of Gaifman and Specker

~

In-

T (0) ~ T ~1 implies

This is analogous

LGa Sp]

that

(in ZFC)

-

there are

2 wq

different

51

-

i s o m o r p h i s m types of A r o n s z a j n trees.

this context we also m e n t i o n that Devlin [De 2] of B a u m g a r t n e r

and proved that,

assuming

morphic A r o n s z a j n

trees w h i c h are

For normal trees

T

automorphisms

T .

on

of length Hence,

~(T)

= ~1

IV.

In fact, Jech [Je 3]

length

o(T) w = o(T)

.

2w

, there are

2 wl

if

T

e(T)

denote the number of

is rigid,

Souslin tree

is either finite,

then

T

has proved that if

c(T)

= 1 , and

constructed T

or

T

with

and

2 ml

~(T)

in Chapter

is a normal

tree of

2 ~ ~ o(T) ~ 2 wl

of arbitrary p r e s c r i b e d

provided

non-iso-

Q-embeddable.

He has also proved that it is consistent

a Souslin tree between

o(T)

has extended a theorem

]R-embeddable but not

wq , let

for the homogeneous

w I , then



In

and

that there is

cardinality

~w = ~ .

Jensen has proved that if we assume

, then there is a homogeneous

Souslin tree

(It is doubtful w h e t h e r this

is p r o v a b l e >+

T

such that

from

Define

.)

We scetch the proof.

6o , M o

chy of models

N'

~(T) ~ w 2 .

Let

(So l a < ~ 1 )

as in the proof of Theorem IV.1.

is defined as follows:

N'

O

A new hierar-

= L 6, [AJ

O

satisfy

where

6'

is

' O

the least

8

countable in case

such that N'

O

A tree

"

lim(a) : An

generic for

L6[AJ T

o-branch

TIo

and

~

ZF- + "8 a is countable".

is defined as before, b

of

b E N' . o

Tlo

on

Tla,

ded to an a u t o m o r p h i s m lim(S),

if

~ E Ms

on

then

a

is extended iff

TI(a+I)

Furthermore,

for

is an extendable

automorphism

on

(This is so b e c a u s e

M s ~ Hwl . )

of automorphisms

T .

Now,

)

and

A .

Let

is

T

on

let

TIS

a

c E Na

N* = L

Let

LIB] [B], ~2

B ~ wI

o < S < w I with

extending

then

M s I=

~"

be a sequence

code up

N v ~ N*,

is

can be exten-

T~m,

(av I ~ < Wl)

M o-

is a normal

We claim that there is an a u t o m o r p h i s m

(~ < ~1 ) , thereby p r o v i n g the theorem. (av I V < ~ l

i.e.

.

"there are u n c o u n t a b l y m a n y automorphisms

on

b

It also follows that if is extendable,

M o is

except for the

It easily follows that

Souslin tree which is homogeneous. an a u t o m o r p h i s m

Thus

o~,

p #o v

T , Nv

be as

-

in the proof of Theorem EC ~ B~ ~,CN~ the closed

~S~. Let

set

induction

on

phism on

TI¥ o

M¥o

be the monotone

a sequence

.

~P~ I v < w1>

such that

enumeration

NyvC My~

: Po

.

is an automor-

a ~(TI¥o)

for

TIyv+ 1

such that

~v+1 o pv

"

PK = ~ < ~

Now let

P~

for

p = U 4) that to

element

in

Souslin

algebra iff

der the reverse

algebra

with

are concerned with Boolean

~

G

such that

is homogeneous

algebras.

if to any o(b)

= b'

iff for all

~Ib

(the algebra obtained by intersecting

b ).

A complete

~

has a dense

ordering.

Boolean

algebra

~

b,b'

E

It b E~, every

is called a

subset which is a Souslin tree un-

(More on this concept

in Chapter VIII.)

-

5}

-

Now it is easy to prove that the canonical complete Boolean algebra constructed from the totally homogeneous Souslin tree of the last paragraph is a homogeneous Souslin algebra with at most Using the trees of Theorem I

and Theorem 3

wI

automorphisms.

we obtain in the same way

a rigid Souslin algebra and a Souslin algebra with exactly 2 phisms. w2

Finally, the definition of the homogeneous Souslin tree with

automorphisms

homogeneous. least

automor-

w2

can be modified so that the tree will be totally

Hence we obtain a homogeneous Souslin algebra with at

automorphisms.

C h a p t e r VI

MARTIN'S

We have

seen e a r l i e r

A X I O M A N D THE C O N S I S T E N C Y OF

that,

given a

are c a r d i n a l a b s o l u t e g e n e r i c

c.t.m.

extensions

M

SH

of

NI

ZFC + GCH

and

N2

of

, there

M

such

that (i)

NI ~

2$ = w 2

&

(ii) ~N2(w) =~M(~)

~ SH

and

N2 F

In this c h a p t e r and s u b s e q u e n t f

e ~Si

GOH

ones we shall show h o w it is p o s s i b l e

I

to o b t a i n m o d e l s

NI

and

N2

w h i c h give

ency r e s u l t s

SH

in place

the c o r r e s p o n d i n g

consistI

for

of

~ SH

.

The c o n s t r u c t i o n

w e l l known,

and there are at l e a s t

two good e x p o s i t i o n s

literature,

so we shall only s k e t c h the proof,

and use

of

N I is

already

in the

this s k e t c h

to

I

motivate

the c o n s t r u c t i o n

in the r e m a i n i n g

of

chapters.

N2 Most

w h i c h we shall give

of the r e s u l t s we shall m e n t i o n here

are due to S o l o v a y and T e n n e n b a u m . first established Suppose

the r e l a t i v e

then we are g i v e n a

a cardinal absolute in

~

.

ticular,

Thus,

in

generic N

any S o u s l i n

w h e n we pass

to

same cardinals, N

N

consistency

c.t.m.

M

linity"

by g a i n i n g an u n c o u n t a b l e

branch)

in

turns

N

tree

SH

ZFC of

M

it was

they who

.

, and we w i s h to find such that

SH

holds

to be no S o u s l i n trees.

~

in

M

antiehain

M

and in

M

N

to have

will

the

still be an

can only lose its "Sous(or, a f o r t i o r i ,

an

~1-

to c o m m e n c e by seeing h o w

the S o u s l i n i t y of single S o u s l i n However,

In par-

to lose its " S o u s l i n i t y "

~1-tree

So it is not u n r e a s o n a b l e

out to be r a t h e r easy.

l a t e r chapters,

of

w i l l have

it is clear that an

, so a S o u s l i n

we can d e s t r o y

of

Since we w i l l r e q u i r e

in

.

M

extension

~1-tree

N

In p a r t i c u l a r ,

there w i l l h a v e tree in

in full d e t a i l

tree in

to s i m p l i f y

M .

This

the n o t a t i o n

this s i m p l i f i c a t i o n w i l l be e x t r e m e l y h e l p f u l . ) ,

(In let

-

us first of all amend Aronszajn

we simply r e p l a c e (chapter II).

~T

than "depth",

and

terms of

rather

that n o w a S o u s l i n Thus,

if

T

force w i t h

tree)

by

To o b t a i n

re-

the a m e n d e d d e f J n i t ~ o n ,

in the o r i g i n a l d e f i n i t i o n s

still

talk about

"height" h o w e v e r ,

this w i l l m e a n than

~T

just what it did before,

")

The a d v a n t a g e

tree is a s p l i t t i n g poset

is a S o u s l i n tree in a

N

of

T , we do not d e s t r o y any cardinals. question,

but in

of this a m e n d m e n t

satisfying

c.t.m.

rather

is

c.c.c.

ZFC

, then if we

It turns out that

for we h a v e

already proved

in

( T h e o r e m II.7)

I N

be a

(Thus, limit b

of an

everywhere

C h a p t e r II the f o l l o w i n g l e m m a

Let

(and h e n c e

~T

we also a n s w e r our o r i g i n a l

Lemma

~1-tree

so that they "grow d o w n w a r d s "

as p r e v i o u s l y .

(We shall

~T

-

our d e f i n i t i o n of an

tree and a S o u s l i n

ther than " u p w a r d s "

55

c.t.m,

of

ZFC

in the real world, ordinal,

and so

T

~

, and let is an

has

if we take a S o u s l i n

be a S o u s l i n

a-tree

for

a-branches.)

Let

T

just in case

b

tree

T

in a

tree in

~ = w~

2~

w i l l be a c o f i n a l b r a n c h of

Hence,

T

e.t.m.

a countable

b c T . is

M

N.

Then

M-generic

of

ZFC

for

and

N

form a g e n e r i c e x t e n s i o n for in

T , then

T

N

N

M[G]

killin~ ly,

~ .

T .

by t a k i n g a set

tree in

We r e f e r

N

extension either:

it is dead and e v e r a f t e r r e m a i n s

that a h o p e f u l m e t h o d

iterate,

in some manner,

one at a time.

tial that the e n t i r e

G

w i l l be,

to this p r o c e s s

N

of

N[G]

once a S o u s l i n dead~

as clear-

, then tree is

This last o b s e r v a t i o n of

SH

w o u l d be

to

1, k i l l i n g all the S o u s l i n trees in

in o r d e r for this to work,

iteration

N-generic

, since

for o b t a i n i n g a m o d e l

lemma

Now,

M[G]

G

This term is j u s t i f i e d because,

any f u r t h e r g e n e r i c

will not be a S o u s l i n

sight,

M

w i l l not be S o u s l i n in

the S o u s l i n tree

killed,

of

, a c o f i n a l b r a n c h of

if we m a k e

means

M[G]

can be "handled",

it is e s s e n -

to some extent,

with-

-

in the original set theory,

Well,

any finite

number

We describe

be much easier their

M :

-

we want

our final model

and the only way we k n o w how

of forcing.

ged.

model

56

in the case of such)

associated

(complete)

is just a disguised this is clearly

not

below.

boolean

with

(and hence

is easily it turns

~ , concerned,

algebras.

form of forcing

irrelevant

this

However,

the posets,

(Since

BA(~)

as far as killing

arran-

out to

but rather

forcing

with

- see chapter

T

of

is by the method

extensions

successively,

this a r r a n g e m e n t

to consider

to do this

of two generic

formed

to be a model

I -

is concerned.)

This

N

will not be apparent next lemma, vantage

nificant

Let

but already

will be quite

to denote

Lemma

in the case

"complete

of two extensions

for doing

~

significant.

boolean

strengthening

"successive

in the

extensions"

the ad-

From now on, we shall use

algebra".

of lemma

described

The f o l l o w i n g

lemma

"BA"

is a sig-

1.8.

2 (Solovay) M

M~Bo )

be a

c.t.m,

be such that

M , a ~ique such that

of (in

m2

~A

M (~°)(]B1)

ZFC M)

.

Let

IB°

be a

I~ I is a BAI~O

BA

= ~

in

.

~ M (]B2) , and a canonical

Let

Then there

domOB 2) -- ~:x~M(]Bo)l

(with

M .

IIx~:~l]Po

complete

]BIE

is, in

= "n:})

embedding

e :]B ° - ]B2 . Furthermore, (i)

if

Go

]BI/G ° is

whenever is

M-generic

~i ) , then there

if that ~o that

G2

for

(the BA in M[Go]

such that (ii)

]Bo, ]BI, ]B2

is

]Bo whose

for

~2

e = idI~ ° in the above) and there MEGo][GI]

is an

GI

name

in the

°

which

is

we have:

M[Go]-generic ]Bo-forcing

is

M-generic

for

language for

~2

. then

(assuming

G o = G 2 0~ o

MEGo]-generic

= MEG2]

as above,

and

G2

is a set

M[Go]KG I] = M[G 2] M-generic

are related

set

GI

for simplicity

is for

M-generic ~i/Go

for

such

-

Proof:

This

of

fix E ~ II~ ° = ~] sense

of

~2

a unique and

such that The rest

2

trees

in one go.

, and,

the m a x i m u m

to define

principle

z E M (~°)

for

llb2 = ~I~ ° = b °

enables

~i)I~ ° = ~

is routine,

us to handle

any finite

That we do not destroy to r e m a i n

nitely

Souslin

trees,

M (~o)

to find

z E dom(~ 2)

element

b 2 E ~2

o

trivial.

number

any cardinals

so when we come

we require

in the

~

but not

I

.

11b2

and

the

x v y

such that

be that unique

e(bo)

describe

= {xEM(~o)

' let

this

|

of Souslin in the process

to deal with

the f o l l o w i n g

infi-

lemma:

3 (Solovay) M, ~ o ' ~I' ~ 2

(in

be related

M)II'~BI has

The obvious

Suppose

now we use lemma

of Souslin (~nl n ~ ) complete

trees

in

of

BA's

idea works.

M .

morph

embeddings

Then

~

~

Thus,

~

to form

We call

~

w

are in [Jell,

system for

(in

I='~ o has

~ '~2 has

with

we construct

is best

an

an

m~

is to regard the direct

union

86.1

~)

of is de-

with

this

as follows,

Iso-

function

~n"

Let •

Lemma

, enm

M) associated

of course.

the c.c.c."

w-sequence

n+1

described

c.c.cJ'

w-sequence

(enm I n ~ m ~

is the identity

algebra ~

M

M

successively

BA

enm

If

, then

(where,

This

is a boolean

(One way

= BA(~) ) .

in

A natural

so that each

= Un~w~ n

completion.

to deal

union.

2.

The details

enm: ~ n - , ~ m

is its direct the system

2

: ~

and a commutative

fined by composition). system

as in lemma

the c.c.c."l~o

Proof:

set

dom~B2)

for example,

of the proof

clearly

Let

simply

of

but for

and

e .

We shall

(in the sense

is obvious,

Let

and

(by normality)

b o E ~o

Lemma

~2

' use

llz = x v y

Given

Lemma

-

is just lemma 85 of [Jel].

construction

many

57

~w

on

be its

as a poset of the chain

and

-

(~n] n < w ) identity, plete

we call

the

embeddings

en

(~Bn)n~ here

In the case where BA

, will

.

the embeddings with

, the direct

Suppose

algebra

-

, together

:~n-~

~ (enm)n<m~)

as the limit

~

58

limit

then we take

(murderous)

were not

the

the c o r r e s p o n d i n g

in our iteration.

all of our previous

enm

of the system

(in If

com-

M) G

•

is

as defined M-generic

work be incorporated

for in

w

M[G]

?

lemma, Lemma Let

The positive

answer

in c o n j u n c t i o n

with

]Bo, ]B2

be

in

BA's

~ .

]~1 i s

in

part

[SoTe]

of lemma

by our next

2.

generic

~

Since

systems

of any

ultrafilters

(limit)

length,

chain bra of

BA's

~8

of the Proof:

such

union

~ ¥ 's .

that

Then

of

cf(~)

Theorem

the canonical ~o

~I

name

is obtained (in

M) for

M-

|

we state

and let

(a)

(~8

~ a

By a combinatorial when

speaking,

is g u a r a n t e e d

construction

ordinal,

' (b) y < a ~ ~ y

is the direct

(in

will work

by the next

equally

it in a fairly

well

general

for

way.

- Tennenbaum)

be any limit of

such that

em-

~ }~(]B2)

any cardinals

limit

be a complete

]B I c M (IBo)

Roughly

on

the above

5 (Solovay ~

M(]Bo)(]B1)

through

does not destroy

lemma.

Lemma

and

~2

e :]Bo-~]B 2

is an element

for details.

by f a c t o r i n g

That

M , and let

Then there

a BAI~O = ~

See

Proof:

Let

the second

is supplied

4 (Solovay)

bedding

M )

to this question

49.

¥ w I .

~ T

iff if

u

E u

the

point

of

.

A partial

map

Let

be

Claim

observe

p

Dx =

[p E P

Let

X = [D x I x E T] X-generic map

in

has

from

say

is s p e c i a l .

u

of

is a n e a t

T

, and

such in

~

into

p.e.'s

of

subtree

that:

T , then

conversely;

an e x t e n s i o n

subtree

on e v e r y

they

and level

Q

is an o r d e r - p r e -

T

into

ordered

Q .

by i n c l u s i o n .

c.c.c.

of this

claim

follows

I x E dom(p)}

set

u

of

of a l l

the t h e o r e m

let

tree

same h e i g h t

from a neat

the p r o o f

preserving

u

satisfies

how

the

(p.e.)

the p o s e t

~

Ignoring

of

embedding

serving ~

have

We

subtree

same h e i g h t

every u

tree.

is a f i n i t e

x,y

have (ii)

Then every Aronszajn

be a n y A r o n s z a j n

N

(i)

an

is a s t r i c t l y

is w e l l - d i s t r i b u t e d ) .

Proof:

of

There

.

Since G

for ~

~

into

for

the

from

it.

Clearly

time b e i n g , For Dx

IXI = w I < 2 w .

Clearly Q , so

~

each

let us x E T ,

is d e n s e , by

f = UG

MA

in there

~. is

is an o r d e r -

is s p e c i a l .

-

66

-

We turn n o w

to the p r o o f of the claim.

countable.

We show that

ible elements. mains

X

Firstly,

must

of

~) .

that if

p,q E X

of

Secondly, and

.

of

~

, y E dom(q) of

(Of course,

for example,

in some a r b i t r a r y ,

dom(p)

p,q

be the n u m b e r

Pick

k ~ n

largest

linearly

u°

If

k = n

is a n e a t

member

of

assume,

of

~

Let

m

k

U

p E X

of

X

U

orderings Since

determines, of the levels

e a c h level

every m e m b e r

of

of S

~]

E U

.

Let

such that

S~ ~

and taking

u

X k

of the do-

T-level unique

for that

(k+1)'st l e v e l

of

. of all dom(p)

section

~ ), a final

is countable,

of

we can assume

(5)

of

S

in

S . such

such that

By l e m m a 7, we can find

is w e l l - d i s t r i b u t e d .

Tm

that

w h i c h do not p r o v i d e

P i c k an a r b i t r a r y m e m b e r

for

our l i n e a r

S

has u n c o u n t a b l y m a n y e x t e n s i o n s

~ = hts((~) )

of

We may

o

of

X

X.

such that e v e r y

( k + 1 ) ' s t level

(k+1)'st l e v e l s

of

of

and that

(eg. u s i n g

Tm

Let

subtree w i t h

in a n a t u r a l way

(Throw away those m e m b e r s nice p o i n t s ~ )

of

be the c a r d i n a l i t y of the X

,

each level

k ~ n

levels

lies on a

be the set of all .

k

, that the

the d o m a i n of e a c h m e m b e r of Let

so a s s u m e

with

to

such

of any m e m b e r

a single n e a t

we are done,

by choice of

member.

ordering

has d o m a i n an e n d - e x t e n s i o n

m a i n of each m e m b e r

dom(g)

so that u n c o u n t a b l y m a n y m e m b e r s

subtree

X

and

these o r d e r i n g s . )

in the d o m a i n

have a d o m a i n w h i c h e n d - e x t e n d s levels.

correspond

but c o h e r e n t manner,

of l e v e l s

assume

, but we can e l i m i n a t e

just those i s o m o r p h i s m s w h i c h p r e s e r v e n

(as n e a t

there m a y be several

for each such pair

this a m b i g u i t y by,

are i s o m o r p h i c

x E dom(p)

then

isomorphisms

that the do-

we can ( e q u a l l y clearly)

the i s o m o r p h i s m

= q(y)

be un-

c o n t a i n a pair of c o m p a t -

X

one a n o t h e r u n d e r p(x)

X c ~

we can c l e a r l y assume

of all of the m e m b e r s

subtrees

Let

Pick extensions

B~a (51) ,

-



of

0]

is a set.

[I P E P o

&

Thus, in

Work in

q E Q & &

]P2 is a neatly

~-closed poset.

thus

(in

f :P2 " ]Pc

cover.

Define

]Pc in

sily seen that h

is the

(in

e

e-basic projection,

= p .

f

e

Clearly,

Clearly, for

to

]Bo

then

h~2

]PI"' so q E Q

of course. f

is a neat

p E]P o , e(p) = uniquely by the it is ea-

And clearly,

= f .

if

The rest of

I

and simplicity)

(hmnln<m<w>

and

=

is a neat cover,

and that, using lemma 2, we constructed <enm I n < m < w } ,

sequence from

is a complete embedding.

the lemma follows easily now.

0Bnl n < w > ,

]Bo of course),

by setting,

Since

Suppose now that (for definiteness

p =

This also proves ~

(and extending ]Bo).

if

using the fact that for each

enw: B n ~ w

projection of

into

fn (p) = Pn "

is a neat cover for

=

In order that

(iii), all that is required

define

Pw

and partially order

= BA(~ )

' condition

condition

; and for condition

tion argument,

We first of all set

iterated forcing, we must ensure

(Since

Pn E ~ n

the fact that the

It is easily verified

is completely embeddable

that

tion

which also adds no new subsets

n E ~ n & Pn =hn+1,n(pn+1))}

Wl-Closed poset.

of all define

--W

Can we define a limit oper-

Wl-happy.

be a suitable limit operation for that each

(Then

enm , hmn ) is al-

In fact we can do so very simply, utilising

entire

system.

Now, by Lemma I, we know that if we force with any of

ation , suitable for iterated forcing, of

~1-happy

en~

is a

en~ -basic

h~n~w

~n'S

= fn "

, but also for

~1-happy,

so that we can

, together with the embed-

en~ , n < ~ , the inverse limit of the system

((~n)n]

q = qwl

obtain

¢(v+I)

¢ : wi+I

If

tri-

and work along

a sequence

function

be a proper

E D v , and set

v ~ w 1, and

define

are

At limit

below in v e r i f y i n g

= max(dom(qo)) qv+1

u-sequence.

cofinality-sequence

it in the same way as d e s c r i b e d

q~+1

in the i n d u c t i o n

such that If

set

lim(~),

~(v) =sup~ 0 ; ~ ;

subalgebra

~

of

such that

= ~ , then

~ ~ , ~

;

= o~B

~o

Lemma)

Assume

, x ,

Then

, 0 ~ j < i ~ 2 , be the basic

be any S o u s l i n i s a t i o n

But clearly,

12

subalge-

9 2 .

T a° = h I o "T aI and ~ E A12 - T ~1 = h21 "T a2 "

Lemma

"

is a nice

•

such

is a that:

-

Proof:

8?

-

VJe c o n s t r u c t the sequence

(]B I v < w 2)

proceed,

we c o n s t r u c t

the f o l l o w i n g

(a)

h

:~

(b)

T~ ~ v

' an a r b i t r a r y

(c)

[~ :~

~ ~ , where

-~

also

(~

are o t h e r w i s e

T)

, the basic

by induction. sequences:

projections;

Souslinisation

~

As we

of

has d o m a i n

•

~I

but

V'

"J

arbitrary;

(~) (e)

C T

(v~)

, an a r b i t r a r y

such that for all Tv : h 'Je let

closed u n b o u n d e d

a 6 C

, ~vI~ = YvOa

set in

, ~[~=

wI

~

,

"T ~




earlier).

O.

.

O

Case

2.

Set

~v+l

Stage

v + I .

= o(~v'(]Bm IT < v>)

Condition

(iv) holds

by lemmal

and induction.

Case

3.

Let

~ = otp(Ax)

meration fine

~X

Stage

of

k , k < w 2 , lim(k)

, and let

AI .

from

Notice

, cf(k)

<X(~) I ~ ,

for some

c E C , y ~ ~X(~)c

Then there is

x' ~ ~k(~)c

~k(T),k(v)(x')

= y

By assumption

~Iv)(y)

(~,x>

and

~k(v)

k(T) (x) >X(T) © e -I

seen that

x

v ~ ~ < 8 and

x ~ yX(~) and y ~)ff~(~),X(~).

such that

ql )/y

lies above

are such that

x' ~k(T)x

and

X(T)(X)) hk(T) , k(~) (~-I Since

.

Hence

c E C , it is easily

~k(~)c (in terms of



as claim 3

is

(~,y , x) E U

= ~k(~(n)),k(~)(yn)

Then

Ta ' for each

Ta

such that

= y , and

with

such that

, is as required.

of

X t

(~,y,x>

For each

, s

9

satisfies

n < ~)

(**)

.

with

• < e .

.

is defined.

•

in

~(n) ~ T , all

a

2.

Tia

x

This is

z .

~(T)

, h k ( v ( n + 1 ) ) , k ( ~ ( n ) ) ( y n + I) = Yn

x n E T~(n)

s(~,y,x>

s

1 and

y ~ (~~ )

n < w , Yn,Xn

E U

claims

, and

< w)



for all

: (i)

To

and

that

the c o m p l e t e n e s s

B ° ~ B I ~ ..., we can pick a n o r m a l

Y

We say that a M

such

that

As before,

5

as in lemma 6

p E T

fixes

Ii

v

then

{~

~ w11A n v ~ w }

contains

B E M .

from lemma

by an argument

for the rest

Ti~

iff there

as in lemmaV]XI.5.|

of this proof.

is a normal

a-tree

t

in

I!

p II-TI~ = t .

for each

that every

set

-

a < wI

p E TB(a)

the closed u n b o u n d e d

there

fixes set

TI~

is a least •

Let

{~ E m I Iv < ~

ordinal

B(m)



- B(v)

(v)

h(<x,f>)

-- x iT x,

&

we shall assume

f muf ,

: x .

In order that our definition does not break down, we ensure that at every stage the following

two conditions hold: o

(*)

If

lim(a)

= sup[f(s)

(**) Let

and Is

(x,f> E ~a+1

> °t}

TX

' then for all

t E (Tx)Ya,

f(t)

.

<x,f> E T a + I , x' ~

<x,f>

and f'(ti) =

ZF

minus the power set axiom.

and

(N

I ~ < w I>

by a simultan-

as follows.

and for all is countable

LsEb,C qa~ (ii)

be such that

such

a

such that

b E W

Ls[

=

T

Na = N

°

Y~+2

[~l~,ml~+2,F~] .

Set

t o : {~

and

~I = {<x,> I X E T y I }

= ~

for convenience).

Suppose define

;

~+I

is defined and

~a+2

by forcing over

(where we have assumed that

~I(a+2) Na+2

"

satisfies

llTo= {0)II~

(*) and (**) .

We

-

Let

S = {<x,<x',f>>ix

For each

~ mya+2 &

105

-

<m

x

s = <x,<x' ,f>> E S , let

mya+1 & <x,,f> ~ ~I(~+2)}

x, e

ps = [P I P

.

is a finite function

&

dQm(p) ~ (T ° & ran(p) ~ Q & (Vt E dom(p))(Vs E (Tx,) (t

Na+2-generic subset of

into

~+2

It is obvious that

Suppose next that (*), (**).

For each

"

Note that

~I(G+3)

x E Tya , let

with

Put

p E Xs, p

( x , f U X s , p> ~

<x',f> .

still satisfies (*) and (**).

lim(a), a < m I , and

We construct

~s

~a

~la

is defined and satisfies

by forcing over

B x = {b I b

~

.

is a cofinal branch of

T°x such

o

that, for some extends

b}

For each ~

<x',f'> ~

Regard

ran(p)~ ~x

as

<x",f">& p2p'.

p x E N~ .

For each

x E T

follows.

Let

Xx, u .

which

.

a poset under

~la

y E Tya+1

Set

and each Xx, u

be an

u E ~x

, we define an element of

Na-generic subset of

fx,u = U{f' I (<x',f'>,~> EXx, u] .

~x

such that

~

as u E

Since (**) holds for

(and to some extent since (*) holds also), it is easily seen that

-

fx,u

is an o r d e r - p r e s e r v i n g

U[p

I (3d)( ~

.

suppose

, where

~.Ve define

For each

Ta+ I

x E T

lim(a), to make

.

TylC

from

?y,x,u(t)

that,

<X,fx,u>

and for each into

that

(y,}y,x,u > ~

~a

b E

Note that

(**) to consider here.

~l(a+1)

a < w I , and

is

defined

(as above).

(*) hold.

' Y Eb] TM[C][b]Ic

&

into

Q

~I

in

. in

Then

fb

M[C][b]

M[C][b]

] .

Clearly,

Tb =

is an orderpreserBut look,

, so, using

C

is a

fb ' we can

-

107

-

o

easily Q .

define,

Hence,

, an o r d e r - e m b e d d i n g

~ "T

speci ll

is

It remains

Le~a

NEC]Eb]

M[C]~b]

WII

second

in

to check

is special"

:

of all of

, whence,by

Tb

choice

into

of

b ,

.

that

~

of which may appear

is Souslin, rather

We require

strange

two lemmas,

the

at first glance.

7

Let

D E }~[C]

Set

~b = {(x,f> E ~ I x E b]

Proof:

be dense

Since

T

Thus,

if

and

D0~b

and where

Let

b

~"

MrC]

that

, b

is

in

~b

x'

in

~b

E T~ here

I~]B "<x',f'>

T .

"

T .

xo E b

&

<x,f)

is over

M[C], ]B

Pick

of

on

we can find

set term for here.

branch

M[C]-generic

the forcing

x = x°

Then

is dense

x o l~B"(x,f>

, where

assume

be a cofinal

D 0 ~b

is the generic

_(~ <x,f)

x' --

<x',f')

in

in

branch

of

b 1,...,b n

T , and set

be cofinal

M'[bl,...,bn]-generic M[C]

p : [bl,...,bn}

as above,

branches

subset

still.

- Q , and set

}~' = M[b]

of

, T ° = U{T x 1

of

T ° , and suppose

WlM

over

~M

, with

Set

~b = [(x,f> E ~ I x Eb] . n ~P = [(x,f> E ~ b I iA1(~qi < Q P ( b i ) ) (

o

VtETxlC)(tEb Then

Proof:

i ~ f(t) < Q q i ) ]

D 0 T~bb is dense

Let

P(bi)

in

T~

.

Let

D E M'[C]

be dense

in

~b

"

.

= r i , i = 1,...,n

~(x,f)~ i = [z E T° I htTo(z)

.

For each

= htT(x)

<x,f>

E ~b

& (3r.~ v

-

.

, we can apply Corollary in

M'

S ~(B) (z) is well-distributed)

we may assume that

.

& ht(

is well-distributed).

-

By Lemma

6, we can find

that for all that

Y~I

a normal

v < w I , Sl0(v)

= al

and

-

function

= S n V0( v

(VS_>~1)(p(ys)

val = 21

Let

a

Now,

Sa+ I J ~ , so by the construction

B > a •

By SyB

If follows,

by

that

< u , and

S

YS

u'

~

E ~b

of

~emma

for some

ht(u')

' u'

, u'

, which

such

~ < ~I

so

~a'

.

SyB+I

nature

D , contrary

the proofs

to

of lemmas

such

for all

8 > a+1

, if

D

is a

B > ~ •

nEu'~

being

u'

of

~,

6 (~b) ,

is well-distrubuted.

B = ht(u') Y~

NyB+I

of the construction

YS

Set

So, as

E ~B

S (~> n [u~

S

(and the first

u = (x,f) ~ u o

E NyB+I

B ) for all

is to say

~a+1

E S~+ I , (~) E ~u~

SyB,

E D .

a > h t ( u o) =

By

= ~ .

will be well-distrituted!

YS

can lie in

on

= B , then

< u

of

(~>

Hence,

induction

and

S °) we can find

SIyB+ I C WyB+1

$<x) ~ ~u'~

By examining

of

(3) and the generic

~

u

Pick

M

So, we have:

y~ = ~ ~ ao,al

E NyB+I

~ [U~

seen that

and,

(4),

(by a simple

Let

that

in the d e f i n i t i o n

ZF--model,

u'

such

u E (~b)a+1

Let

EVJ0(v)

in

(vBz~I)(SlYs(wyB)

be least

clause

o : w I ~ ~I

= YB ) "

(4)

that

&

109

dense

(2),

(~>

But we have

Hence in

E

just

no extension

~b "

|

4 to 8, we obtain

9

The conclusions

of lemmas

4 to 8

able

model

ZF- + "there

transitive

At last we can prove:

of

are valid w h e n e v e r

M

is an uncountable

is a countcardinal"

+

110-

-

Lemma

10

M[C~

I= "~

Proof:

is a S o u s l i n

Let

J

be

(J(a,~)

Let

X

there

I a E0n)

is a set

is c h o s e n

Let

To

be

set

TI

be l e a s t

such

lemma

we

the l e a s t

3

, and

setting

a = aN

(i)

~ , TI(~+I)

(ii)

C O a

(iii)

there

is

N(T)

= Tim

dom(w')

and

dom(N')

of

NKC 0 a ]

each

Tim

.

apply b x}

~I~

•

x E T a , set

Now, lemma

Tim 7

, to c o n c l u d e

that

that

the

,

Since

~,X

E M[C]

,

E L[Y,C]

and

(X,~>

.

T,T,W

We can as-

E L[Y]

= J(To, ~ I ~ set

E

~'(~)=

a maximal

I (BxEXa)(y~x)]

b x = [y E TIa I X < T Y ]

N[C 0 ~]

~

'

is a S o u s l i n to

set

, N ~H M , INI M = w , such that w2 t r a n s i t i v e set U E M , such that,

D = [yE~la subset

any

N E M

over

a] ~

, w(W)

Set

is a d e n s e

For



~ H M[C] w2 = X a = D e f X 0 (~la)

w'(X)

chain

that

~ = NN

N[CN

that for

.

~,X

w I = w~ [Y]

such

~'

M[C]

that

,

Ciearly,

and

in

, such

a countable

is a m a p

1i"'-1:

and

~

ordinal

U-generic

such

L[~]

of

so that

obtain

and

function

enumerates

Y ~ Wl, Y E M

that

< Y , T o , T I) E N

~I

antichain

sume

Applying

.

the u s u a l

be a m a x i m a l

Y

tree"

D N Tbx

is d e n s e

in

Tbx

of

can

I x'

, each x E T a.

-

Again,

for each

x E T

, let

111

-

Bx

be the set of all b r a n c h e s

b

of

o

Tx

of the form

w E (Ty)a

.

b = [z E T °y l w < T o z]

Let

arbitrary.

bl,...,b n E B x

By choice

of

for some

, and let

U , C qa

is

y E T~(a+1),

y ]

, where, f o r

each

~ ; ~i+I (p) = (ai(P)'Pi+1>

, define

p

Co(p),...,an(p)

"

similarly.

Recall now the proof of lemma VII.2. For each

n < ~ Set

'

let

~P# n

=

[P= I (Vi - Vl

such that

h(q2) = ~2"

On(

for some

finish by setting p~ E]P*n "

Since

qience from

p2

v1 = f(pO)

]P* n '

h(Pn+1) .

=

We may

pl E ]P* n

'

(in C) &

Let

v2=f(plo ).

Similarly, pick p2 E ]P* 2 - @.

fiX ~ ~I × ~II~ ~ = ~ v = ~,~

Then,

, so by construction,

IIX is speciall~* Thus

for

II~ ~ ~I × ~II~* Aronszajn

tree

= ~ , of

IIEvery Aronszajn tree |

Appendix ITERATED In [Jn Jo], Jensen constructible forcing.

and Johnsbr&ten

ing over

Jensen developed L

~(w)

give a generic

construction

of a non-

w-iteration

of Souslin

of an

is a specialisation

of an earlier

in order to show that iterated

must introduce

and because

cumbersome,

FORCING AND

A I3 set of integers by means

Their construction

tion which

der,

SOUSLIN

new reals.

the construction

Souslin

For the convenience

in [Jn Jo]

which

forc-

of the rea-

is (necessarily)

we outline here a simplification

construc-

fairly

just gives the above

result. In

L,

we define

a sequence

(i)

T°

is Souslin;

(ii)

if

bo

is an

define

(iii)

~I of

branch TI

of

in

Lib o]

if

is an

Lib o] - generic branch

define

a subtree

bI

is Souslin

if

(bo,b~,...)

C

of

T2

~I

is Sous-

~I , then we can cano-

in

L[bo,b I]

such that

L[bo,b q] ;

is any such sequence

T°,T I,...,

then

regardless

of branches

L[