Languages with Expressions of Infinite Length

LANGUAGES WITH EXPRESSIONS OF INFINITE LENGTH

CAROL

KARP

Associate Professor of Mathematics University of Maryland

1964

NORTH-HOLLAND P U B L I S H I N G COMPANY AMSTERDAM

8 1964 North-Holland Publishing Company N o part of this book may be reproduced i n any form by print, microfilm or any other means without written permission from the publishev

PRINTED IN THE NETHERLANDS

PREFACE

1956 x =0

x =2

x =1

go

:

no

1957 ;

by on

1956. on

on

1960.

VI

1958,

1.

1960 a,p

do a

p

1962. ;

As on,

on

1960. by

31, 1963 University of Maryland College Park, Maryland

FOREWARD O N SET T H E O R Y

no

(ZI

book [41].

[6]

A y # z.

( x , y ) , ( x , z)

f

f(x).

{{x} { x , y}}.

(x,

f

S

flS f 0g

f(g(x))

xE

n S.

f

f, g

x T

T

g(x) E SF

T,

B S = {x: x C

S y ES

US = { x :

x

f

f-1

x E y}.

ns = { x : x E y

yE

U(Sg: i E

u sc. i€I

on I

ZI{Sr: i E i E 1.

f(i)E Sc x

x, y E

x Ey

x =y

by

< E”

6 8

6 E E. U6

E

xC

y EX. 6, E 6 s(6) = 6 u (S}. s(U6).

“6 E

8

XI1

FOREWARD ON SET THEORY

n

+ 1 = (0, . . .,n}.

o,

as S+O=S

+

+

6 S ( E ) = s(6 E) ~ + E = U ( ~ + E : ~ < E } 6.0 = 0 = (8.E) 8

+

B a s ( & )

c5-c = U(6.t :

5 < E}

A A

of

6

“

5 <E

E

SO

XC.

“>”

A <x>

0 x; on.

1 <x y> = ((0, x )

< 6).

x)).

on

A U(S6: f

<x> =

7

on on 7

all

S

6 < 6. S

8

w = 00.

og

$.,

all wa

CL.

CL+ og+

= og+l.

will

< 8.

00,

CL

/?=

8e

CL

XI11

a


: 5 < W ( ^ < < E ( 6 0):5' < Dom(E) - 6)) El6 = ^ LwI

0

0

< yn

<w

< n < w.

n

< w,

Ano. . . A n v . . .] Aoo.. . A o v . . .I].

A consistent

6

[A A Ao...At.. [ A A At] t[A + [+it]] I- [A+ [A .. [A A

A

6,

. .] by5.2.4,

Ao. . . A t . . .]I.

Ao,.. .] A

vi A,,] A n + l , o . . .An+l,,.. .I], vn+l

all t,

At]]

Ao. . . A t . . .]]I

t- [A+

Aoo. . .Aov. . .I]. Aoo. . . vo AOvo all i < n [AovoA . . . A;. I-A; t)[ A ; A

< yt

< yn+l

[A;

T = {An,,,:n < w}.

A

5.4 Completeness of the Basic Formal Systems with Chang's Distributive Laws of

will a

> w1. A

5.1.3, [

v

Ir= l[A A o . . . A t . . .]IT.

by A

v

= I[Ao A = I[Ao v

58

REPRESENTATION THEORY FOR BOOLEAN ALGEBRAS

on 'B(@a(Z)(La); r)= .

IAIr

I[$

A

< IA'Ir

I-rA + A',

I[$

[+]lr,

v [+]lr.

< lAIlr Y < 6, t-r[C + A t ] t-r[C + [ A A o . . . A t . . .I], l[A A o . . . A t . . . ] I F I[V A o . . . A t . . . ] I T . @a(C)(L,) t-rA I'lkA t)A'.

I[AAo.. . A t . . .]lr

E IT,

by y

Td

=

d

1

E Z.

y

I^Eu Eo # 4, E , # 4. CO [, Eo [XvEb, E , EL]. Ei = E, 0 < v < u. A0 = ^<E:"C,: Y < a) "Ei. A0 = Ai, = ^<Ei"CL: v < u)^Ei. = A = [XvAi,]. [XvAb] = ^<E,."C: : v < u)^E,. A' A = A'.


Fabon A, E Y

g on Y A , V(s,[ x v A ] )= V(s,[ X ~ ' O ~ S F ~ " ~ ~ ~ ( ~ ) A ] A) no yEYn bound by {g(y)). g on Y , V(s,A ) = V(s,S B r A ) E g on Y A. Y ,g d = { A : V(s,A ) = V(s,SBFA)

97

sE

do A). A =A A [Ao!PA1] [ @ A o . .. A t . . .] At EA, A A SB; s on A = [xv Ao], A0 E A , no A. =[x~'O~SF,Y"~"~'~~S byB 9.3.4 ~YAO] V(s,S B r A ) = V(s,[ x ~ S S B ~ A O ] ) . sB,t = t DRngfv). Ao E A , V(sV,t,Ao) = V(sB,t,SBTAo) sv,t E V(s,k v S B f A 0 1 ) = V(s,A ) A E A. 11.2, (***) (***)

8.2.3, 8.2.4, (D 0

9.3.6 Definition. A

12. S) substitution property

R

FaDon Q(x,v) = Q(x, v ' )

v, v'

j3, (*)

s,

Y

ES

j3, (**) sE

S

E

B

Y E S.

1E

D,

on (*) D

(**).

on

D

on y,

D on

by




9.3.10 Theorem.

g (***)

sE y

B

E

on Y

g

B,

9.3.5.

no

y bound by {g(y)}

Yn V(s,[XVBI) = V(s,[xg'OvSF,Y nRng(v)BI).

B' = SF: nRw(v)B,sv, t = Sg'0v.t' E DRng(O'Ov).

t E DRng@)

on x

V

(1) V ( S , = Q ( x ) { v ( s v , t , B ) :~ v , t S} (2) V ( S [Xg'OvB']) , = Q(x){V(sg,ov,p, B ' ) :S

n F V ( B ) n Y ,X 1 =

X O= t , t' t(x)

x

Xo.

paired

S}.

g ~ O v E, ~ ~

t'(x) = t(x)

S, s ~

-

n FV(B) Y . Xi,t'g(x) = , ~ ~ , ~ , t , t'

xE

99

V(sv,t,B ) = V ( S , , ~B’). ~,~, 9.3.7 V(S,~,,,,,B’) = V(s’,B) s’ s,,,~,, on Y A s , , ~ ~ , ~ X O , s’(x) = S , . ~ ~ , , ( ~ ( X )= ) t’g(x) = t ( x ) = sv,t(z).

-

s’(x) = S = FV(B)

FV(B)

s‘ 9.1.5.

= sv,t(x).

V(sv,1, B ) by

, ~ ~ ~ , ~= ( tX’ ()x )

= t ( x ) = s,,t(x).

s‘(x) = ~ , . ~ ~ , ~= , ( s(x) x)

on FV(B).

sv,t

V(s’,B ) =

t, t’ on

will t‘

sv,t E S

t’

t, t’ t 0gl on

s , , ~ ~E , ~

7s

s v ,t

t on

t, t‘

-

E

0v )

on 9.3.9

T 1 sE S ,

E S.

E

s , , ~ ~ E, ~ , by

9.3.9

A

9.3.8

g, y.

g-1,

9.4 Function Notation for Substitution will

v

A.

“A(v)”

“ A ( v ( 0 ) .. . v ( E ) . . .)”. all A and A. f A (f 0v ) = A (fv(0). . .fv(E). . .) bound

fv@)

A

by

a([).

fv(5).

A

bound by

Y bound A:Y = do

A. v

< u}.

g(y0)

2

-

Y 2 A

all

Z, :0

Y

< v}.

< u, g

g(yy)

100

SUBSTITUTION

on Y, SF~w(")SBFA.

A. no

SBrA

v(E) bound by FV(fv(E)). A FV(SBFA). 9.4.1 Theorem. A(f 0v )

A(f 0 v ) = any SBFA FV(A) = fv(f)

A

(*)

(***)

9.3.5.

V ( s ,A ( f 0 v ) ) = V(s',A @ ) ) on F V ( A ) . A

9.4.2 Theorem. A(f 0v ) V(s,A(f 0 v ) ) = V(s',A ( v ) ) on F V ( A ) . = A(v))

by 9.2.10, 9.3.5, 9.3.10.

9.2.10

all s'

E

s' E

< j?, V(s,A ( f 0 v ) ) re-

CHAPTER 10

INFINITARY PREDICATE LANGUAGES

10.1 The (a,p, o, n)-Predicate Languages a

B

< B < a.

/?

0

o

a /3 (a,/?,o,

/I a. FaPo=

n Labon

a 1,

+,

V.

A,

(a,B, m,

Lab

8.1.1 10.1.1 Induction Princifile for Formulas.

A ELI, [ - A ] E A , Ao, A1 E A , [ A o + A11 E A , )

all

y,

s*[yTo.. . T e . . . ]

O(y)((s*To...s*Te.. .>)

=

0

< 5 < 0,

y,

V on S x Faflon

To, . . ., Tq, . . . ES, (3) V(s,[ T O P T I ]= ) R(P)(<s*Tos*Tl>) P, (4) V(s,[QTo.. . T t . . . ] ) = R(Q)(<s*To.. .s*Tq.. .>) Q, Ao, . . . , A t , . . . v

O 0, T, [ T O Y T I ] [VTO ...Tt ...I Tt E U 1 < v}. >0 =K 1 < v, v

< o}

o = K.

K

(4)

o

A

j3


l y,

O(Y)()= TOE do, T I d l

[ T O W TEI ]T, I<xo>l tp,

.

(<do. . de. . .>) =

Te E dE

I To. . .T E .. .] I

[pl To.

. .Te. . .] E T,

l<xo>[

r q-

114

I N F I N IT A R Y P R E D I C A T E L A N G U A G ES

TOE do, T1 E d l [ T ~ P TE ~A ]n r. R(Q)(<do.. .de.. .>) = 1 Tc E dE [ Q T o . ..Tt.. .] E A n I'. R(P)(<dodl>) =1

R(=)

E

EZ

Td

=

d' 1

=

D)(L)) =

B B. 6.4.2

@,(lIpa)

B

Z

7

B(@,&'.; D)(L))

< \All

lAol

k[Ao + A l l .

by I[VvA(v)]l < \ A ( /0 v)I

bound

92

I[VvA(v)]I ] A ( /0 v ) l .

bound

11.3.3 Theorem.

py+#7y;Qy)

y

y

E =

E

D,

< B, y

: (y+, B,

A @y+p(17y; Qy)(L). Dy)(L)) T A 10.3.4. y.

L

93 =

do X d

230

!-A 1

0,

n)-

/I) =y

l[+l]l

y

'B

y

on

h,

T = {C:hlCI

=

1)

10.3.5

[-A]

T

130

INFINITARY PREDICATE

A.

[-A] go

10.3.5 no d:

A y *

-

. . ., Cv(gv 0v,), .

.

* *

Co(g0 0vo), g,

a,

[vv,cY(vY)l,

X

on

5' # v do [Wv,Cr(vc)] 5 < v. W, = 0 v,) + [Wv.C,(vy)]]. 1[lA] A [A W O .. .W,. . . ] I # 0 8. 0.

ki[[iA]

A

[A W o . . .Wv..

.I].

all

1 A I- [V [lWO].. .[ l W , ] . . .I g, 0v,

W,

W,

/?

w,

:E

U

5 < v.

Wa

/?

d

< v}

W..

4 k [V [vgo o vo[lWoll...[ 3 w [ v g Y 0v,[-rWPl11-. .I vV) A [-1Wv,C,(v,)1],

t)

do

v.

I- [vgv 0 Vv[iWv]]t)[[vgv0vrCv(g9 0%)I v < y by 11.2.5 11.2.5

FA. \ [ - , A ]A [A W O .. .W,. h :

80

A

.. ] I

[ivv,cv(Vv)]]

kA. # 0.

/[-A ] A [A W O . .. W,. . .]I

by 1

131

BASIC FORMAL SYSTEMS WITH CHANG'S D I S T R I B U T I V E LAWS

v{-~I[Tc=T;]l:t< ( } v I[qTo...Tc ...I = [CUT T i . . . ] I [ y To. . .Tg. . .], [ y Ti,.. .T ; . . .] T. 1 by 6 9 3 . v \[QTi,.. .TL.. .]I To. . .Tc. . .I, [Q Tb. . .T i . . .]

V { ~ ( [ T cTk]l:5 < q) v (-I[QTo.. .TE...]I) all A.

1 by 695. < 6) = \[A A o . . . A t . .

{IAc(:5 A.

y

y

hlC( = 11, h [-A] E F F 10.3.5 [T =TI E r by 891. A0 = [To = Tb] A1 = [ T I = Ti] r, A2 = [ [ T o ~ J T I ][TbyTi]]E r (A01 A \All IA2/ by 6 9 2 h 10.3.5

F

=


R(Qt)(v) = 1 S on y y.

y

12.3.7

(*)K

K

< y.

12.3.7

12.3.5,

%wW/a u { '332.

~

~

CH,

Z

P(Laid Y )

'332.

12.4 Non-deducibilityof the Axiom of Dependent Choices in Systems with Distributive Laws and Rules of Independent Choices

DCH, V R o ( l I 2 a ; GI,,,)

C tl

y < p. a(@)=




L:,

c)

[VxoVxl[[C(xo)A C ( x l ) ] --+ xo = X I ] ] C(x0). LL, E,

L,

E

by

E

[Vxo[xo

A

. . ., xn-l

xo.

L, R , = { ( d o . . .dn-l> : ( d o . . dn-l> Fm' = ,

L:,

=

Fm. p

by

[ ~ x o .. .x n - l y l j 5 [ Q . ' ~ x o .-.~ n - 1 ]t)C(XO,.

-

- 3

xn-1, y ) ] ]

[Wxo. . .x n - l p y C ( x o ,

. . ., xn--l, y ) ] ] [vxo. . . x n - i y z [ [ C ( x o , . . ., x n - I , y ) A C ( X O ,. . ., xn-1, 41 + y = z ] ] Fm X O , . . ., xn-l, y C(x0, . . ., xn-l, y ) . L,

L:,

0,

q.~n, e

Fm'

=

( d o . . .dn-le>

(D,0 u {(~nO,>},

R),

L:,

A 9"

A. OC

( d o . . .dn-l> C(x0,

. . ., xn-l, y ) ,

A ) = Vm(s,A,) s D, A, C(x0, . . ., xn-l, y )

Fm.

13.2 The Definability of Fundamental Notions of Set Theory L,

(o,

-

=, E, a

%a =

( T a E>

Ta a,

on T a

D

155

THEORY

on

Ta a = y+,

By $ Ta. --

y ETa

13.1.1

13.2.1 Definition. y = xoxl 13.2.2 Definition. y = [xg x XI] 13.2.3 Definition. y [xoTxl]

y = (xo,XI). y = xo x X I . y XI.

xg

13.2.4 Definition. n XO. . .xn-l] -13.2.5 Definition. y = 13.2.6 Definition. y 13.2.7 Definition. y

y

y = {xo, . . ., xn-l}. y

xo

xo.

-

=0 --

y

=

4. y

=

xo.

13.2.8 Definition. 13.2.9 Definition. 13.2.10 Definition.

xo xo

xo

--

13.2.11 Definition. y = [ x o 2 x 1 ]

xo

x1 E

13.2.12 13.2.13 13.2.14 13.2.15

Definition. Definition. Definition. Defination.

C x1 -

xg = E

13.2.16 Definition. -

- -

y =

xo

xg

y = xo(x1)

4 xo

xg

xo 1

xg

) s(v) E x ~ ( v= ) x o [ x 4 ( ~) xolU Rng(x4l~) v = U v E Dom(x4)

XB(S(V))

x4) =

+ 1,

+ 1.

4.

:

13.3.15 Definition.

-x5 =

~ 0 x 4

x5

x4).

13.3.16 Lemma.

A , x1

xo

on

x2

SFFA,

SF(x0, X I , x 2 ) X = ( x : g(x) E X I ) f = SF(x0, X I , x2 ) +. SF

x2

x1

0g on X .

:

g 0 SFTA = n<x5(V)^x2(x0(x4(V))) : v

< (T>^x~(u)

x4 0

XI),

=

x5 =

13.3.17 Definition.

x4).

x3

Z SFxoxlx2

x3

= SF(x0, X l r x 2 ) .

13.3.18 Theorem. L 9 1, 9 2 , V X , , 9V%,, o < y 9VX.,.

V%,.

9 2 , W%y 9VX,.. A = [[VvAo] + SF,R"p@)A0], Ao xE bound by FVf(x).

92

no

T

< a,

x

E

FV(T)

x

163

D E F I N A B I L I T Y O F T H E FORMAL SYSTEMS

T. FVTxo t)

x1

-

01 A

92

- -

Zb Ib x1 A

-

--

rb A

-

XI +

E x3 A E

xo

- --

92(xo) t)

XS+

E

by

A

- &F

EB A

x3 A

x1

5

A - -

+

/? = u

0A xz A

A

Z

FV(T)

t)

E

1 ~ x 2A

-

-

XZFVT[X~~~[X~~~X~]]X

1E

x

B

9%Sy

A

. .[

=[A C