ZERMELO-FRAENKEL SET THEORY A ND CUMUL A TIV E
TYPE THEORY (*)
John L. POLLOCK
1. Introduction. I t is we ll kn o wn th a t Zermelo-Fraenkel set
theory (ZF) is clo se ly connected wit h cu mu la tive typ e th e o ry
(CT). Th e sets o f ZF (wh e n ZF is taken t o include the a xio m
of re g u la rity and the axiom scheme of replacement) can be a rranged in t o a tra n sfin ite h ie ra rch y o f cu mu la tive types. Th e
purpose o f th is paper is to construct a natural axiomatization
of CT and compare it wit h ZF. Ord in a l numbers are taken as
p rimitive , along wit h the ordering relation " < " , and there is a
p rimitive re la tio n symbol T(a,X) ( " X is o f type a"). A fe w obvious a xio ms a re adopted f o r ordinals, together wit h a xio ms
characterizing types wh ich make types cumulative. The axiom
of extensionality is adopted f o r sets, together wit h the a xio m
scheme of comprehension for each type level. I t follows t rivia lly from the above that CT is contained in ZF. But the converse
only holds wh e n th e a xio m o f choice is added. I n particular,
the axiom scheme o f replacement is not a theorem of CT. This
suggests the construction of a slig th ly weakened version of ZF
that is equivalent to CT. I n this connection it is urged that, in
the context o f CT, the a xio m scheme o f replacement is unacceptable. Ho we ve r, i t ca n b e replaced i n Z F b y another in tu itive ly acceptable a xio m — the a xio m scheme o f ordinal re placement, wh ich results fro m re strictio n the a xio m scheme of
replacement t o sets o f ordinals (th e functional image o f a set
of ordinals is a set). Th is a xio m scheme, although weaker than
the p rin cip le o f replacement, s t ill suffices f o r a ll th e uses t o
which the principle of replacement is cu sto ma rily put. Furthermore, th e p rin cip le o f replacement becomes a theorem wh e n
the a xio m o f choice is added. This version o f ZF is equivalent
to CT, and it seems to me that it is in t u it ive ly preferable to the
(*) Th is paper was presented a t t he meet ing o f the As s oc iation f o r Sy mbolic Logic i n Clev eland, Ma y , 1969.
ZERMELO-FRAENKEL S E T THE O RY
4
5
3
customary version of ZF. I t yields the customary version when
the axiom of choice is added.
2. Th e a xio m o f ZE. Th e re a re several versions o f ZF, re sulting f ro m slig h t ly d iffe re n t choices o f a xio ms. Th e re a re
also slig h t ly d iffe re n t versions depending upon wh e th e r sets
of in d ivid u a ls a re allowed, o r, a s has become co mmo n co n temporary practice, o n ly sets of sets. To me, it seems ridiculous
to d isa llo w sets o f in d ivid u a ls. Th u s ve rsio n s o f Z F and CT
which countenance sets o f in d ivid u a ls w i l l be developed here.
However, th is makes n o difference to th e ma in re su lt o f th is
discussion, wh ich is the connection between ZF and CT. Th is
connection s t ill holds i f sets o f in d ivid u a ls a re disallowed i n
both ZF and CT.
As th e variables o f o u r th e o ry range o ve r both in d ivid u a ls
and sets, we must have a p rimit ive predicate " C" meaning " is
a set". We will fo llo w the common practice of using lo we r case
variables a s general variables, a n d u p p e r case va ria b le s a s
shorthand fo r the relativization of quantifiers to sets. Given this,
the axioms are the fo llo win g :
(Z1) EXTENSIONAL ITY: ( X ) ( V Y)EX Y = (V z)(zcX
zEY)1.
(Z2) SEPARATON: I f cpz is a fo rmu la in wh ich z is free but X
and Y are not, then the fo llo win g is an axiom:
( X ) ( g Y ) ( z ) N e Y ( c r z & zeX)1.
(Z3) UNI O N: ( V X) (HY) ( z )[ z E Y ( H W ) (WEX & zEW)1.
(Z4) P OWE R SET: ( V X) (HY)( V z) (zEY = z X ) .
(Z5) P A IR: ( V x)( V y)(HZ)( V w)[wEZ ( w x v w y ) ) .
(Z6) I NFI NI TY : (UX)REY)(YEX & ( V z)zitY)
& ( V Y) (YEX D YUTYIEX)1.
(Z7) REGULARITY: ( V X)[X D (ay)(yEX
& ( V z)(zcy zo,X))1.
(Z8) I NDI V I DUA L S : (E X )(V y)(yE X — C ( y ) )
&( V x)( V y)(xEy D C ( y ) ) .
(Z9) REPLACEMENT: I f cpxy is a fo rmu la in wh ich x and y are
free but z, X , and Y are not, then the fo llo win g is an axiom:
454
J
O
H
N
( X ) [( V X) (XEX (
3
p xOF
y ) CHOICE
A! Xy I)Oc M
D
L. PO LLO CK
(EY)( V
y)
( y E
The theory resulting from Z1 -YZ8 will be called ZF
w i l l be called ZF
Iresulting fro m Z1, Z3, Z4, Z6 -( Z9,
R
2 Ttheorems
.are
h e
t ofhZ,F2.
e ZF
o r y x )
. Z
ZIO
Z
5
3
i to
s2 Z1t ah en d
( x
2
accoalingly
Z,F
t h e o r y
E X
.t Zi shF o f a
2
axiom
choice,
t o r ZF
&
axiom
:t1
i es ofschoice.
t hu e l
r1 h e
c
it sso ss tt
m
p
tf 3.ooCurmu
p
pp lauutive
a e th e o ry. Cu mu la tive typ e th e o ry results
o l typ
x
rlo
m a libre ra lizin g the structureyof simple type theorem. Acco rd from
vw
i
a
d rr type
ds
ing toee simple
theory,
e ve) ry set has a unique type, and the
a
s
o
iknn oo f ag set mu st a lwa ys b e o f typ e one less th a n th e
imembers
I
t
n
w
i a t set. Th e rationale often g ive n f o r t h is is th a t i t
type
o f th
.
to
h
gives
a "constructive"
p ictu re of sets as being b u ilt up b y suctfj
u steps
h fro m objects o f type zero (individuals). However,
cessive
s
Z
e
t
this constructive vie w of sets does not re a lly ju stify the struci
F
fo f simple typ e theory. Rather, i t ju stifie s th e less re stricture
y
w
i ctu re o f cu mu la t ive t y p e th e o ry. T h e co n stru ctive
tive
stru
tpicture
r h requires that each set be b u ilt up out o f sets that have
a
o
n ubeen constructed, but there is no reason wh y these sets
already
tshould
s
f be o n ly o f the n e xt lo we r typ e — th e y can be o f a n y
ti wenr type. Th is leads to o u r defining types b y saying that a
lo
i is
h
t of a given type iff its members are a ll of lo we r type. This
set
e
has the effect o f making types cumulative — if a set is o f one
r
type,
then it is o f any higher type. The lo we st type o f a set is
called
e
its rank.
c Simple typ e th e o ry generally requires a ll types to be finite.
u again, th is is an a rtificia l re strictio n given the constructive
But
r
picture
of sets. There is no reason why, after having constructed
s o f a ll fin ite types, these sets cannot be collected together
sets
i
into
a n e w se t o f t yp e co. Consequently, a cu mu la tive t yp e
o
theory
with transfinite types will be developed here.
n Cumulative typ e th e o ry suffers p h ilo so p h ica lly fro m h a vin g
, presuppose t h e o rd in a l n u mb e rs ra t h e r th a n co n stru ctin g
to
a
n
d
(Z10)
ZERMELO-FRAENKEL S E T THE O RY
4
5
5
them as in ZF. Th is is undesirable f o r the simp le reason that
set t h e o ry sh o u ld id e a lly p ro vid e a foundation f o r o rd in a l
number theory. In this respect, ZF is preferable to CT. However,
this unfortunate aspect o f C T is la rg e ly re ctifie d o n ce i t is
seen h o w ZF and CT compare. Th e p ro o f o f the equivalence
of some versions o f ZF and CT can be regarded as a kin d o f
completeness theorem fo r ZF in that it shows that ZF gives us
all t h e stru ctu re o f CT with o u t presupposing o rd in a l number
theory, a n d a t th e same t ime ca n b e regarded as a kin d o f
soundness theorem for CT in that all the ordinal axioms presupposed b y CT are theorems of ZF.
The axioms of CT can be divided into three classes — ordinal
axioms, t yp e axioms, and set axioms. Th e language o f CT is
different f ro m th a t o f ZF. I t n o lo n g e r contains the p rimit ive
predicate " C" (wh ich can be defined in CT), b u t does contain
a binary relation symbol " T" (" T(a ,X)" means " X is of type a"),
a p rimitive predicate " O n " (" is an o rd in a l" ), th e constant " 0 " ,
and the b in a ry relation symbol " <" . To simp lif y the notation,
we w i l l adopt the abbreviation o f using sma ll Greek letters as
variables ranging o ve r ordinals. W e w i l l also writ e " T
a abbreviation f o r " T(a ,x)" . W e ma ke th e f o llo win g d e fin ian
tions:
x" a s
DEFINITION:a 13 (ci <13va = (3).
DEFINITION: y = a +1 ( V ( 3 ) ( a < f 3
DEFINITION: y = a + 2 ( g ( i ) ( t i = a + 1 & y = (i +1).
DEFINITION: L(a) ( a * 0 & P O ) a — (3+1).
We adopt the fo llo win g ordinal axioms f o r CT:
(01) On(0).
(02) (Va )(V1 3 )[a <fl - - ( f 3 < a ) i •
(
a <y].
0
(04) (V a )(V 1 3 )(a •
3
,
(05)
( V a ) ( a * 0 D 0 <a ).
) f3 v
5
(06) I f gm is a fo rmu la in wh ich a is free but (3 is not, then the
(
fo llo win g is an axiom:
V
(
a
3
)
u
(
)
V
(
f
P
3
a
)
(
(
H
V
a
456
J
O
H
N
L . PO LLO CK
(8) I f 4:03y is a formula in wh ich 13 and y are free but a and ô are
not, then the fo llo win g is an axiom:
( OE( (
3 <61],
)[(
01 - 06 say that the ordinals are we ll ordered b y " < " , and 0 is
3
the smallest ordinal. 06 is equivalent to the p rin cip le o f trans<a
finite induction. 07 tells us that there is no limit to the sequence
( l !
of limit ordinals. 08 tells us that the functional image of a bounY )
ded set o f ordinals is bounded, o r equivalently, a sequence o f
W 1
ordinals o f bounded length is bounded. These axioms g ive us
3
at least a large part of the algebraic structure of the ordinals.
Y )
Types
are characterized by the fo llo win g axioms:
(
g
(Ti) a = 0 D (To ( = (V y)(yex D (a(3)[(3 <a & T5y])).
(T2)b( x ) ( T
)
o
(T3) ( x ) ( g a ) T
(y )
X
a
The
yx . g( inxd ivid
) u a ls are objects o f type zero, so we define:
3
.
DEFINITION:
C(x)
)
Then
( once a g a in w e u se u p p e r case va ria b le s a s va ria b le s
ranging
o n ly o ve r sets.
Y
)
Finally, we adopt t wo set axioms:
[
(S1)(EXTENSIONAL ITY: ( V X)( V Y)[X = Y = (V z)(zeX
1 zEY)J.
(S2)3COMPREHENSION: I f Ty is a fo rmu la in wh ich y is free
< but X and a are not, then the fo llo win g is an a xio m:
a ( V a)(3X)( V y)[yeX = (Ty & Tay)].
S
Let
t us ca ll the th e o ry resulting f ro m the above axioms CT
CTI T
1 is
it
. Psu fficie n t f o r th e development o f mo st o f set th e o ry u p
through
o rd in a l a rith me tic.
i A
s The fo rm the a xio m scheme o f comprehension takes in CT
interesting. Russell's p a ra d o x sh o ws t h a t t h e u n re stricte d
ais 1
axiom
scheme o f comprehension i s inconsistent, a n d y e t i t
r
seems
in t u it ive ly self-evident. In CT
e
1
h a v e
a w e
a
s
"o c o n s t r u c t i v e
"n
a
b
l
y
s
ZERMELO-FRAENKEL S E T THE O RY
4
5
7
form of the a xio m scheme of comprehension, and it is plausible
to suppose th a t i t is re a lly th is f o rm o f the a xio m th a t is in tu itive ly self-evident — we a re misunderstanding th e unrestricted a xio m wh e n we t h in k th a t i t is self-evident. Wh a t is
surely self-evident is that given any formula cp, there is a set of
all individuals satisfying cp. Furthermore, g ive n any fo rmu la cp,
at any stage in the process o f constructing a ll sets (i.e., a t any
type le ve l) we can f o rm the set o f a ll p re vio u sly constructed
sets satisfying cr, and this w i l l be a set o f the next type level.
In o th e r words, g ive n a n y p re vio u sly established d o ma in o f
objects, we can fo rm the set o f a ll those objects satisfying cp.
The problem wit h the a xio m scheme o f comprehension comes
from supposing that we can ta lk about the set of a il sets satisfying cp and that this w i l l be another set o f the same group o f
sets that we began with.
From axioms 06 and T3 it fo llo ws that every set has a lowest
type — its rank. Let us define:
DEFINITION: T * x -= ( (3 ) (T5x = a •-•<•••(3).
Cle a rly there can be no universal set in CT
set1 would contain elements of a ll types, and so could not itse lf
have
, ba e
type.
c aHo
u we
s ve
e r, i t sfo llo
u ws cimme
h d ia te ly fro m the a xio m
of acomprehension that f o r each o rd in a l a, there is a universal
set U
e
THEOREM
I : ( V a) (EX) ( y ) ( y E X T Πy ) .
,
cIn oparticular, there is a set U
n
so isf a t rivia
x It
a l l l matter to derive a ll of the axioms of ZF
iCT
s
i n1 dw i i vt hi id n u a l s .
tI i
THEOREM
2: A l l theorems o f ZF
n
. g
1
a r e
t h e o r e m s
o
C converse of theorem 2 is not possible, fo r the simple reason
The
o
fothat thef languageCof CT
e
a
of CT
ntherefore
1 i s r there
i c hare
e theorems
r
rl1 ti h •a t
st ZF
of
h a na r e
l1
n
o
ahre sentences
t e
o f Z.F
ethat
t
t.qwhether
e
1
a
v
r
e
this
e
is
n
the
case,
but I conjecture that it is.
l a
n
g
u
h
oe r t e em sn c e
utsa h ege n
W
ieo
s
f
f
n
nZ i F.
m
F
g
t1
g
sh
l., t
o
,yIa
n
fh
o
n
'
:d
tot
yw
k
n
458
J
O
H
N
L. PO LLO CK
The ordinals are presupposed by CT
be Iconstructed in the usual manner. For example, we can define:
. H o w e Ord
v e(x)
r , [ ( V y)(yEx D y c x) & (V y)( z )
DEFINITION:
t
h
e
y
[(yEx & zex) D (yEz v zey)1 & C(x)].
c
a
n
Leta u s ulse b osld fa ce
o le tte rs a s va ria b le s ra n g in g o v e r t h e
constructed ordinals. We can then proceed to develop o rd in a l
number theory just as is customarily done in Z.F. Unfortunately,
we q u ickly encounter a d ifficu lty. Th e p rin cip le o f transfinite
recursion cannot b e ju stifie d i n C T
finitions
I
b y transfinite recursion, we must proceed as fo llo ws.
First,
we
. I n define:
o r d e r
tDEFINITION:
o
I f (tpxy is a fo rmu la in wh ich x and y a re free,
j u s t i then
f y
d
e
cr[x] =
y such that ozpxy;
0 otherwise.
/ the
y such that cpxy if there is a unique
Then we mu st p ro ve the fo llo win g " re cu rsio n p rin cip le " :
Given a n y fo rmu la clay and o rd in a l x, there is a unique
function F such that domain (F) = x & ( V y)(y < x D F(y) =
(f[F Y
)
This ]cannot
be p ro ve n in C T
1 .it cannot be proven in ZF
that
I, f oo rn e constructs t h e se t o f functions F y satisfying t h e
ciple,
theorem
.p Ir ne cf p
oi r rsy <oex ,vl ayin dn then
g
defines F = y < x ) . Un f o rttunately,
hh i ne eCT
y1rs < xa
e} nexists.
ac d u mInr ZFs ethei customary
o
remedy is to add the a xio m
scheme
r . Fe of replacement.
n
a
s
oThen if we define 15((y,f) to mean " f is a
Z
function
n
p
r
&
d
i
o
ma
in
n
(f
)
y & ( V z)(z<y D f(z) = iTi[f z])", w e
1
-o
have
N
y
l
F
n
e
yc{O [ y]
,a ; y<nx} exists. The recursion principle follows immediately
a
n othis.
d t However, in CT there is a simpler and more obvious
nfrom
b
yr (wh ich w i l l in tu rn suggest an alternative re me d y f o r
premedy
toZ.F). h v
In C T
e
e
' x and the presupposed ordinals. CT
ordinals
a
t
h
determining
However, it is natural
g oi ev te s whether
u
sthese arenthe same.
o
ia' w
h
a
y
o
f
m
tw
a
sh
c
v e
h
e
e e
m
s
t
o
e
ft w
r{ o e
k l
p
F
a i c
ZERMELO-FRAENKEL S E T THE O RY
4
5
9
to add a n a xio m t o th e e ffe ct th a t th e presupposed o rd in a ls
and the constructed ordinals are the same things. This removes
some o f the a ir o f myste ry f ro m presupposing the ordinals in
order to construct sets. I t has the effect o f saying that we are
not re a lly presupposing the ordinals at all, but rather constructing them side by side wit h other sets. A ll we are presupposing
is some o f th e algebraic properties th a t w i l l re su lt f ro m th is
construction. Furthermore, th e addition o f th is a xio m has the
unexpected side effect o f a llo win g us t o p ro ve th e re cu rsio n
principle, and hence ju st if y definition b y transfinite recursion.
Let us add the axiom:
CORRESPONDANCE A X I O M:
( V x)(Ord(x) —= On (x)) & ( V a)( V ti)(a<t3 a o ) .
Let CT,
! An
b eimmediate consequence o f the correspondence a xio m is
t characterization
h
a
o f the ra n k o f ordinals. I t fo llo ws b y transe
finite
induction th a t a fin ite o rd in a l n is o f ra n k n + I a n d an
infinite
ordinal a is of rank a:
r e
s u
THEOREM
3: I
œr-1
c
t
l t i
2
Tn T g ( V a )
From this together wit h T i and T3 it fo llo ws that e ve ry set o f
[ordinals
t ( a < tisobounded:
D
h
THEOREM
4: I
Te
*
T
V
ao T (
( 3 < a ) J.
X
)r )
Proof:
X
is
bounded
by the ra n k of X.
[&y (
V
(. We ju
o st if y the p rin cip le o f transfinite re cu rsio n b y p ro vin g
a
special
case of the prin ciple of replacement. Notice that in the
x) )
abcve
proof
(• x •E X o f the recursion p rin cip le we o n ly need the p rin D
•ciple of• replacement fo r cases in wh ich the set whose image is
O
abeing formed is a set of ordinals, i.e., we o n ly need:
If (fay is a fo rmu la in wh ich a and y are free but X and Y are
n
D
not, then
('
X
P
( V)) X)[( V a)(aEX D (3!y)(pay) D (HY) ( V y)(yEY ( H a ) (aEX &
)a
D
]• ( P .
(.Although
H the general principle of replacement is not (apparentaM ] . )
(
V
f
3
)
(
f
3
E
X
D
,
460
J
O
H
N
L. PO LLO CK
ly) a theorem of CT
2cement" is a theorem. Th is theorem results fro m the fact that,
in
, ordinal
t h i s a xio m 08, we have in effect an even more restricted
rcase
e so f t the
r i p crintcip le o f replacement — the case in wh ich both
ethedset whose image is being formed and the image are sets o f
pordinals.
r i Suppose
n c
X is a set o f ordinals and [pay is a fo rmu la
iassigning
p l ea unique object to each element of X. Let 0at3 be the
formula
"(ay)(cpay & T y ) . Then to each ordinal in X, 0 assigns
o
f
a unique o rd in a l — th e ra n k o f th e o b je ct assigned t o th a t
"
o
r
ordinal b y cf. B y theorem 4, X has a n upper bound. Th e n b y
d
i
n
08, the ordinals assigned by 0 to members of X are a ll less than
a
l
some o rd in a l y, i.e., the ranks o f the objects assigned b y cp to
r
e
elements o f X a re a ll less th a n y. B u t then b y th e a xio m o f
pcomprehension,
l
these objects can be collected into a set. Thus
a
THEOREM 5: I f yay is a fo rmu la in wh ich a and y are free but
X and Y are not, then
c T2( V X)
R V a) (aEX D l O r a y ) D (HY) ( y ) ( y E Y
(act)(aEX & yay))i,
The a b o ve p rin cip le o f o rd in a l replacement su ffice s f o r t h e
justification of definition b y transfinite recursion. A n examination o f th e use o f th e general a xio m scheme o f replacement
in Z,F
2
proven
wit h its help, i t can a lwa ys be replaced in CT
r e b y oft ordinal
2
principle
h e replacement, together wit h other principles
v e
sanctioned
b y CT
a(because
2
l
i t does n o t contain the general p rin cip le o f replacement),
.s T h CuT s ,
of
2
t Zl F t h o u
a
2
n
eh v CT
e
h Let
g
.rC
t3h eT l
enough, th e general p rin cip le o f replacement is
aInterestingly
e
b
s se
of CT
ta theorem
2
ci3 t oh
Theorem
according t o wh ic h e ve ry se t is e q u ip o lle n t t o a n
ordinal:
n
s. eTt h e
a
a r xi e ei o m
w
THEOREM
6:
1-1
n
lo s s ukf
a
e 1– ( X )(g a )(Hf ) f:ct O
nto > X.
a
c
l
h
t
o
rl
ltoi o c
eh
la
p
fi f m
tnl a i
e
h
s d
Z
e
t d
E
c2
th i
u
he n
seN g u
ZERMELO-FRAENKEL S E T THE O RY
4
6
1
We ca n n o w p ro ve th e general p rin cip le o f replacement a s
follows. Suppose ( V x)(xcX D (aly)cpxy). B y t h e n u me ra tio n
-1
theorem, fo r some a and f, : a onto > X. Let 0(3y be "cp(f((3),y)".
( 13) (13Ea D (a!y)0(3y), so b y th e p rin cip le o f o rd in a l replacement, ( y ; (3(3)(13ca & illiy)} e xists. B u t t h is i s ju s t th e se t ( y ;
(3x)(xcX & cpxy)). So we have:
THEOREM 7: I f cpxy is a formula in wh ich x and y are free but
X and Y a re not, then:
( X ) [ ( x ) t x e X D (H!y)cpxyl D (3Y)( y ) [ y E Y
CT
(Rx)(xeX & cpxy)fi.
This theorem, together wit h the fact that CT, contains a ll the
theorems o f Z,F
ZB
1
2, The
s h faoctwthat
s the general p rin cip le o f replacement does n o t
,seem
t h toabe ta theorem of CT. provides what seems to be a good
areason
C
Tf o r suspecting th a t if th e a xio m o f choice is n o t true,
n3
then the replacement principle w i l l not hold. Speaking loosely,
dcthe aoxio n
m o ft choice imp lie s th a t o u r h ie ra rch y o f sets is n o t
ha
i
n
s ta ll. In other words, it te lls us that if we lo o k at
as fat as it is
eU
a , the set of a ll sets o f some type a, and consider a functional
nrelation
l
l wh ich ma y map the elements o f U„ onto new sets of
cp
ctsuccessively higher types ("standing U. on end" so to speak),
eh
there w i l l be an upper bound (i to the ranks o f the sets in the
aimage
e
of U . In other words, the situation looks as fo llo ws:
lThe
t
h
reason
fo r th is is that the a xio m o f choice puts an upper
se
oon the "fatness" of each U
bound
t oe an ordinal. B u t with o u t the a xio m o f choice there re o
r
tlent
m b y the
s i pm
,omains
o ssib
p lility
y iofnhaving
g
sets fa tte r than any ordinal, in
o h case
might be no upper bound to the ranks o f the
tfwhich
a there
t
f
iZ
sets
in tthe c f
B
icannot
- i m a gthemselves
se
be collected together into a single set. Th is
3is
suspicious
o f the presence o f the a xio m
e
o a freason
q
uU f oir being
p
o
l
o f replacement in Z.F2.
-.scheme
„ ,
a
n
d
h
e
n
c
e
t
h
o
s
e
s
e
t
462
J
O
H
N
L. PO LLO CK
u
c
,
Ul
u
o
4. Ze r me l o - Fr a e n k e l Set Th e o r y . I t has b e e n s een t h a t C T
1
contains
ZF
converse
inclusions. I t is e a sily shown that Z I
I
2
This
n s beCtaken
T in2the •straightforward sense that
, ac inclusion
no n
d t a icannot
all
o f C L are theorems o f ZF
C theorems
T
2
that
the language o f C L contains p rimit ive symbols n o t con3
tained
o f ZE . But we can translate sentences
,c fo o nrin the
t language
h e
of
C
L
th
a
t
contain
typ
e symb o ls in t o sentences o f Z F
st ai im p l e
2
then
rn seshow
a th a
s t g ive
o n na n y theorem o f CT
,Z a .n do f ZF
2
theorem
for
,F i the
t ssake
t rofacompleteness.
n s l a t i o n
2
i.2 The
ordinal
s
numbers
can
a be constructed in ZF
T h i s
problem
tra n sla tin g th e predicate " O n " , t h e re la tio n symb o l
ia 2 s
" , somo
r the
h e r e" 0 " in t io th se language o f Z F
fn <,a
i tl constant
problem
id 2na r ois th e typ e re la tio n " T" . Letting U. be th e set o f a ll
objects
o the
n sets
l yU. are characterized by the fo llo win g
g
Z. Tr Fhofoetype a,
two
principles:
u
n
d
3
,. (i) U
b
N (ii)
0 a 0 D U. = U{U13; ii< a ) U g (Uo ; (3<a))
u
o --tw Il (
w
e x
it ;
lu C
ls (
g
e x
o
x )
ZERMELO-FRAENKEL S E T THE O RY
4
6
3
(where g (X ) is the p o we r set o f X). Consequently, these sets
can be defined in ZP
2 bcayn be defined b y stipulating th a t T(a,X) x c U
"T''
tarivia
r a l nmaf tte
i nr to
i t vee rif y th a t a ll the a xio ms o f C T
T3
are
.r e
2
eI xtctheorems
c uei pr st s o f ZE).
f aoThe rproof of T3 is mo re involved, but
i o bendone
can
. as fo llo ws. Fro m the a xio m o f re g u la rity we can
obtain
what
Ta rski calls the principle of set-theoretic induction
T
h
(which
is
e
n a theorem of Z F
1
tTHEOREM
h 8: ( x ) [ ( y ) ( y E x D c.py) D Tx] D ( x)cp x.
)e:
ZF
1
r
e
l
The
p
rin
cip
le
o
f
set-theoretic induction is a ctu a lly equivalent
a
t
i
to
th
e
a
xio
m
o
f
re
g u la rity. Ne xt, u sin g the a xio m scheme o f
o
n
replacement, we prove:
THEOREM 9: ( y ) ( y e x D (E a )T
ZF2
a
y ) D Suppose
(
Proof:
e ve ry me mb e r o f x has a typ e . Th e n e ve ry
3member has a unique lowest type (its rank), and consequently
aby) th
T e p rin cip le o f replacement there is a set X consisting o f
a
all ranks of elements of x. Let a be the smallest ordinal greater
xthan
. a ll the ordinals in X. Then T
a T3 follows immediately from theorems 8 and 9.
x . Consequently, Z F
CT3.
2 But we saw that CT
3 ca ol ns ot a i n s
CT
strengths
versions o f ZF and CT thus f a r conc C
3
o nT t oaf the
i n different
s
sidered:
Z 2P
a
r3 .
ZF
. I
e
t
1
T fBecause
e
h o e lofr the
l e naturalness
f
of CT
o
q
o
we ,s f o r the less n a tu ra l b u t type-free a xio ms o f ZP3.
justification
3r 3
Z
u
. "constructive'
Ont, the
t hC
ih s F c a n
i3of
a
. bp
choice,
i cTet tuthreere seems t o b e n o questions b u t th a t Z F
vasound
Z
o c i sfo1
3
theory.
nF
an s Unfortunately,
t r u e dthe situation is n o t quite so nice
dwith
a
s 3
a e respect
- t ssto, ZF
lthe
o
i ca a xio
2
3f m o f choice, b u t as w e have seen, there is reason to
ebe
n
ae
about
th e a cce p ta b ility o f th e a xio m scheme o f
.o Zskeptical
P Cnt
n2
i
n
s
a
ci T s c
e
ttp h t e s
2
tm
t
o Z hs
t
hp
e
o F p
u
ela a
2 rx
osi
Zoe
rtm
P
it
h3
ee
.o
464
J
O
H
N
L. PO LLO CK
replacement when one does n o t have the a xio m o f choice. I t
was re ma rke d th a t C T
2
principle
o f replacement, is adequate f o r the d e riva tio n o f a ll
, wtheorems
the
h i c h o f Z.F
reason
d t oh ato
2
e thope
s th a t a somewhat weaker version o f ZF can be
found
still type-free, w i l l be equivalent to CT
n e owhich,
p
o p t l although
e
2
In
c effect,
g
eo n nsuch
e t r a th
a e o ry can be obtained b y simp ly replacing
the
.a
i
laxiom
n l y scheme o f replacement in ZF
2
of
ordinal
b
ya h replacement.
t nh e
Mo re precisely, le t us define Z,F
t
w
a
4
the
t
x
t
o
h
e
i
o
ry
b
o
t
h
e
m
a
t
re
su
lt
s
f ro m a d d in g t h e a x io m sch e me o f
te
.
sT
ordinal
c e replacement
m to eZF
g
hh ne
1
ordinal
ie
r replacement
s
a
l (wh ich is a theorem o f C T
.g W
2
the
ju stifica
e i tio
h n ao f vtransfinite
e
recursion. Ad d in g this a xio m to
ZF
)sv s eu f ee
f i c ne s
f
o
r
1
Furthermore,
we
can
p
ro
ve
a
ll
th e a xio ms o f C T
ts
h
a
t
,t o t hh e r e t h a n
2
T
tw
need
pi
rsome
i additional
n
c
imachinery fo r its proof.
je
p
l
e
DEFINITION:
I f cpxy is a fo rmu la in wh ich x and y a re free,
co
u
then X is closed under cp if f (V x)( y ) [ ( x c X &
sa
f
g a y ) D y EX1.
tn
d
a
The closure o f a set A
se
o u n d e r
A
ifexists, but not so in ZF o r CT. This is because the closure, if it
c
o
p
i
s
i
n
t
a
existed,
h mig
e h t contain sets o f a ll ranks, wh ich is impossible.
n
Z
However,
s m a although
n
l l e th e closure does n o t necessarily e xist as
.e
sa set,
d
t i t is s t ill possible t o define a predicate wh ich charactterizes t h e elements o f th e closure. I n n a ive se t th e o ry, t h e
F
cs
e
h
2
ltclosure o f A
.e
o
of
c theo function
n
tF defined by transfinite recursion by specifying:
tu n d e r
T
sa
i
n
i
n
yc (i)
h
F(0 ) = A
g p
e
p
e
0a F(ct+1 ) F ( a ) U 6 / , (3Y)[1r g F(a) & cf,Yyj);
cd (ii)
e
p
(iii)
; L (a ) D F(a) = U (F((3); p <a ).
n
u
rsb
n
In
ye ZF and CT no such function F exists (where a function is a
o
d
set),
because its domain would be the set of a ll ordinals, wh ich
m
o
d
e
f
e
does
n o t e xist . B u t b y t h e f a milia r th e o re ms t h a t j u s t i f y
fb
ir n
e
o
d
c
fla
p
ss
T
.
jt
3
I
iu
h
n
se
n
rtu
a
a
n
i
ts
ZERNIELO-FRAENKEL S E T THE O RY
4
6
5
definition b y tra n sfin ite re cu rsio n , w e ca n instead d e fin e a
formula Fx y such that
(i) A
o F[a+11 = K a ] U f y ; (3Y)(Y c
(ii)
; L (a ) D Fla] = U {F[13]; [3<a);
(iii)
&
cpYYI);
provided t h a t f o r each se t X , (y; (3 Y)(Y c X & cpYy)) e xists.
Then to say that the union of the range of F is closed under cp
means
( X) ( [ (
X ) ( X E X D PP) xEF I 1 3 1 ) & c p X y l D ( 3 ( 3 ) yEM [ 3 ] ) .
That this always holds is the p rin cip le of set-theoretic closure.
It can be proven in ZF
I a s
THEOREM 10: I t is a theorem of ZF
f o l l o w s :
1 t h a t
i •-f= (
(
V
tions
(i) - (iii) above, then
X ) ( 3
3
1
Z
( X) ) ( [ ( x ) ( x E X D (3(3)xeFt13l) & cpXyl
/ ) (
(( ZV
z
)
F
z g3 E
Y
Proof: Fo r e a chX
a
,
l
e
t
X , , = X n L I M N ; ti<ctI). L e t
0
K = ( X ; X„ 0 1&
.
K
e
xists
b y th e p o we r se t and separation
)
axioms. X = UK.IY K is we ll ordered b y th e re la tio n R = {(X „,
X0); a < t i & X„EK
& X0EK1. L e t a b e t h e o rd in a l o f t h is w e l l
T
E
Z
R
ordering. Th e n there
is fu n ctio n f : a l
zi
ont
formula "f(b) = X0".
By ordinal replacement, {
)n
l
This
N ; is 6a <
setao f}1•ordinals,
e x so
i sit is
t bounded
s .
b y some o rd in a l fi.
o
>
K .
L e t
But this means that
fo r each y, X. c F[fi]. Thus X c F[(3). Then
,
Iif cpXy,
l b yEF[(3+1].
t i
b
e
a
t From
h the p e
rin cip
n le of set-theoretic closure we obtain:
d ( V x)[(Vy)(yEx D ( ( 3 ) T y ) D (HtirroXl.
THEOREM 11. I—
ZP
F
Proof: Let cpxy bes ywhich
g exists
x " b.y athe power set axiom. Thus F can be defined as
T
above,
h w
e i t hn t h et re su lt t h a t U „ = F[a +1 ], a n d L (a )D Fla] =
z1.1{Uoi (3<a). Th ui s b y t h e clo su re p rin cip le , le t t in g y = X,
;[(V x)(xEX D ( s
f
(3 3 Z )
(( 3 )ZX E. U p ) i
e
g&
s
X
X
c
&
g
o
cX
f I
l
n
z( 3 )
)
d
=( 3 )
i
X E 1 .
466
J
O
H
N
L . PO LLO CK
The p rin cip le o f set-theoretic in d u ctio n (th e o re m 8 ) i s a
theorem o f ZF
1
theorem
11 we imme d ia te ly obtain T3. Thus Z 1
to
, CT
4
ai sn de q u i v a l e n t
2
h Consequently,
e n c
we have a type-free theory equivalent to C T
.e 2 1this th e o ry suffices f o r a ll the uses to wh ich 2 F
and
2o irily
ma
s fput.
c uFurthermore,
s t o adding th e a xio m o f choice t o t h is
Z F gives us ZF
theory
4 t o the mo re customary Z F
3
ble
., T o
2
a tatee Un iveZrsity
m
.S
F of Ne w Y o rk at Buffalo J o h n L. POLLOCK
n
4
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m
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m
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s
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e
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o
e
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e
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r
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i
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