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Chapter 1.2 Axioms
CHAPTER
1.2
Axioms
The axiom who f o r m u l a t e d p l e s discussed can be deduced
1.2.1
system t o be i n t r o d u c e d i n t h i s s e c t i o n i s due t o P. Bernays i t i n 1961 (see Bernays 1976) i n s p i r e d by r e f l e c t i o n p r i n c i b y Montague and LPvy (LPvy 1960). A l l theorems i n t h i s book from i t .
GENERAL AXIOMS,
We f i r s t d i s c u s s a s e t o f axioms which can be j u s t i f i e d by t h e conside r a t i o n s s e t f o r t h i n t h e p r e v i o u s s e c t i o n . A l a r g e p a r t o f s e t t h e o r y can be developed from them. 1.2.1.1
I M P R E D I C A T I V E A X I O M OF C L A S S S P E C I F I C A T I O N ,
T h i s axiom has been proposed by s e v e r a l people. The f i r s t mentions of i t seems t o be by Q u i n e ( Q u i n e 19511, Morse (Morse 19651, K e l l e y ( K e l l e y 1955, Appendix) and Mostowski (Mostowski 1950). I t a s s e r t s t h e e x i s t e n c e o f a c l a s s t h a t c o n t a i n s a l l s e t s s a t i s f y i n g a g i v e n c o n d i t i o n we m i g h t f o r m u l a t e t h i s axiom by:
Fa& w a y p m p e n t y P thane A a d a b s A calzclidfincj
t h a t P (XI,
04
th.e
hcth
x nuch
However, t h i s f o r m u l a t i o n where P i s a v a r i a b l e r e f e r r i n g t o p r o p e r t i e s . i s i n second-order l o g i c , because i t c o n t a i n s q u a n t i f i c a t i o n over propert i e s . Thus, we have t o r e p l a c e t h i s axiom b y f i r s t - o r d e r axioms.
A n a t u r a l method f o r t h i s purpose, i s t o r e p l a c e p r o p e r t i e s by f i r s t o r d e r formulas. T h i s procedure was proposed by Skolem and Fraenckel
.
I n o r d e r t o do t h i s , we have t o d e f i n e c l e a r l y t h e l o g i c a l f i r s t - o r d e r language. A l l axioms w i l l be formulated i n t h e p r i m i t i v e language e which P w i l l be c o n s t i t u t e d as f o l l o w s :
A ) V a r i a b l e s . V a r i a b l e s a r e lower-case and c a p i t a l s c r i p t l e t t e r s w i t h o r w i t h o u t s u b s c r i p t . For i n s t a n c e A , x 0,
....
B) Constants.
-
( i ) L o g i c a l constants: 1 ( t h e n e g a t i o n symbol), + ( i m p l i c a t i o n symb o l ) , V ( d i s j u n c t i o n symbol), A ( c o n j u n c t i o n symbol), (equivalence symbol), V (universal quantifier), 3 (existential quantifier), 3 ! (the q u a n t i f i e r ' t h e r e e x i s t s e x a c t l y one'), = ( t h e i d e n t i t y symbol), and (,) parentheses).
5
6
ROLAND0 C H U A Q U I
(ii) N o n l o g i c a l constants. The b i n a r y pred c a t e An e x p / r e h i o n o f
eP
E
( t h e membership
r e l a t i o n symbo ).
i s a f i n i t e sequence o f symbols.
Among t h e ex-
p r e s s i ons, we d i s t i n g u i s h t h e formulas: (1) I f x , y a r e v a r i a b l e s , t h e n
x
=
q and x € y a r e formulas.
( 2 ) I f J, and 9 a r e formulas, and x i s a v a r i a b l e , then 1 9 , ( 9 + $ ) ( @ " G), f@-* $1, (9 s)@'+ Wx 9, 3 x 9 , and I ! x 9 a r e formulas. ( 3 ) A l l formulas o f L
o f (1) and ( 2 ) .
P
,
a r e o b t a i n e d by a f i n i t e number o f a p p l i c a t i o n
Greek l e t t e r s w i l l denote expressions.
9 , J,, 0 w i l l be formulas.
Bound and f r e e v a r i a b l e s a r e d e f i n e d as usual. Sentences a r e formul a s w i t h o u t f r e e v a r i a b l e s . We say t h a t t h e f o r m u l a J, i s a ( u n i v e r s a l ) W xn 9. c l o s u r e o f t h e formula 9 , i f J, i s a sentence o f t h e form W xo
...
A l l axioms, d e f i n i t i o n s , and theorems o f o u r t h e o r i e s a r e sentences. When a s s e r t i n g such sentences t h e i n i t i a l s t r i n g o f u n i v e r s a l q u a n t i f i e r s w i l l be g e n e r a l l y o m i t t e d . I n o t h e r words, when a s s e r t i n g a c l o s u r e o f 9 , we s h a l l u s u a l l y w r i t e 9 . I n w r i t i n g formulas, some parentheses w i l l be g e n e r a l l y o m i t t e d . Those o m i t t e d should be r e s t o r e d as f o l l o w s : F i r s t proceed from l e f t t o r i g h t and when a q u a n t i f i e r i s reached, we a s s i g n t o i t t h e s m a l l e s t p o s s i b l e scope. We r e p e a t t h i s process f i r s t f o r 1 , t h e n f o r A and V , t h e n f o r +, G X [ y ] is t h e formula o b t a i n e d from 9 by s u b s t i t u t and, f i n a l l y f o r f-r i n g a l l f r e e occurrences o f x by Y. L a t e r we s h a l l i n t r o d u c e e x t e n s i o n s o f L by adding new n o n l o g i c a l P symbol s.
.
Our f i r s t axiom, o r r a t h e r axiom-schema i s t h e f o l l o w i n g : AxClass (Schema), Fon m y 6ohmuf.u 9 0 6 I: which d o u n o t contain A P @ee, any d o ~ w r e06 ,the 6oUowing dotunLLea A an axiom: 3 A tlx(xEA-@~A 3 U x E U ) .
Note t h a t t h i s schema (as a l l schemata) r e p r e s e n t s an i n f i n i t e number @ may c o n t a i n any o t h e r f r e e vao f axioms, one f o r each f o r m u l a @ o f L P' r i a b l e s besides x ; t h e o n l y f r e e v a r i a b l e excluded i s A. '3 U x E U ' expresses ' x i s a s e t ' .
1.2.1.2
AXIOM OF EXTENSIONALITY
8
T h i s axiom s t a t e s t h e e s s e n t i a l p r o p e r t y o f c l a s s e s as extensions o f p r o p e r t i e s , namely, t h a t c l a s s e s a r e determined b y t h e i r elements.
AXIOMATIC S E T T H E O R Y
7
The converse implication i s a l s o t r u e by v i r t u e of the laws of logic.
T h i s axiom excludes Urelemente, s i n c e i t implies t h a t t h e r e i s j u s t one o b j e c t with no elements, namely, t h e empty c l a s s .
AXIOM OF SUBSETS,
1.2.1.3
This axiom expresses t h e f a c t t h a t our universe i s closed under ' s e t of . . . I where instead of we p u t any element of t h e universe, i.e. any s e t . T h u s , a subclass of a s e t should be a s e t .
...
3 U bell A Wx(xEa+ xEb) + 3 U aeU.
Ax Sub. 1.2.1.4
AXIOM OF REFLECTION,
This fourth axiom enbodies t h e p r i n c i p l e t h a t i f we have a universe Y already given, then we can take V as a s e t of Urelemente in order t o form another universe V . T h u s , V will be a s e t in t h i s new universe V ' and any c l a s s A (subclass of V ) will a l s o be a s e t i n V ' .
...',
We assumed V t o be closed under 'element of i.e. any element o f an element of V i s an element of V . Thus, I/ a s a s e t i n V ' should have t h i s same property. Before introducing t h e Axiom of Reflection, we need some notation. For each @ of I: t h a t does not contain U, will be the formula obtained by P r e l a t i v i z i n g each bound v a r i a b l e X t o the formula V q ( y E X --* y E U ) (i.e. t o t h e formula t h a t says t h a t X i s a subclass of U). This formula will be abbreviated by X 5 U. More p r e c i s e l y , @u i s defined .by recursion a s follows: ( i ) If @ i s x E y ( i i ) If @ i s 0 vel y.
or
-,dJ
x = y
or 10
( i i i ) I f @ i s WX 0
, then
(iv) If @ i s 3 X 0
, then
@'
, then
, then
@
U
@
U
is @ itself.
i s O U + dJu or 1 OU, U
is
W X(X c -U
is
3X(X c - U A 0').
+
respecti-
0 ).
All other logical connectives can be defined in terms of these, so we do not need t o include them i n the d e f i n i t i o n . Notice t h a t i f @ i s 3 ! X 0 , then @ U i s equivalent t o 3 ! X(X c - U A 0'). The axiom schema of r e f l e c t i o n i s , then, AxRef. (Schema). b and containn at W A(@ YEA
+
m0b.t
3 u( 3 U
A y E u ) +@:[
bl
Fox each am& @ 06 I: t h a t doen n o t c o n t a i n u o h P A Q~e-e,t h e 6o&eowing 0 an axiom: uEU
1.
A W
X W q ( x ~ q ~ ux e u ) -+
A W b( W q(yEb
++
8
R O L A N D 0 CHUAOUI
T h i s axiom says t h a t i f A i s a c l a s s t h a t s a t i s f i e s t h e p r o p e r t y dJ, then t h e r e i s a s e t u, which i s t r a n s i t i v e (i.e. X E U i m p l i e s x C - u), such t h a t t h e common p a r t o f A and u, i.e. A n u , s a t i s f i e s dJu. The j u s t i f i c a t i o n o f Ax Ref based on t h e i n f o r m a l n o t i o n s runs as f o l Suppose we have a c l a s s A p o s e s i i n g a p r o p e r t y dJ, i.e. such t h a t I ; t h i s c l a s s i s a subclass o f our u n i v e r s e Y . I t i s c l e a r t h a t @[A] V i s e q u i v a l e n t t o dJ [ A ] , because a l l c l a s s e s a r e subclasses o f V . We now t a k e V and form another u n i v e r s e V ' such t h a t V E V ' . Then, i n t h i s new universe, V i s a s e t u. Since V i s c l o s e d under 'element o f . . . I , u satisfies:
lows. 9 [A
x W y(x€q€u Also, A c - u. i s t h e same as A . V as dJ [ A ] .
B.
Thus, i f b s a t i s f i e s
v
-+
XEU)
y(qEb
Therefore, s i n c e V i s u , dJ:[bl
.
-
Y E A A ~ E u ) ,t h e n i t
expresses t h e same f a c t
The t h e o r y o b t a i n e d w i t h t h e axioms o f t h i s s e c t i o n w i l l be denotedby 1.2.2
L I M I T I N G AXIOMS,
The f o l l o w i n g axioms exciude c e r t a i n c l a s s e s and make t h e development o f t h e t h e o r y simpler. However, t h e y a r e n o t as w e l l j u s t i f i e d as t h e prev i o u s ones. These new axioms have been e x t e n s i v e l y by s t u d i e d metamathem a t i c a l l y , e s p e c i a l l y w i t h r e g a r d t o q u e s t i o n s about t h e i r c o n s i s t e n c y and independence. Thus, i t i s convenient t o deal s e p a r a t e l y w i t h them. 1.2.2.1
AXIOM OF GLOBAL CHOICE,
T h i s axiom excludes s e t s and c l a s s e s t h a t cannot be w e l l ordered (see 2.2.3.21 f o r a d e f i n i t i o n o f w e l l o r d e r i n g s ) . Ax GC.
v x tj y ( x , q E A A x # q 3 z Z E XA 1 3 z ( z E x A z E y ) ) 3 8 V x ( x E A -, 3! ~ ( q E A x y E B ) ) A 3 C( W U ( W x ( x E U x E C ) A 3 x XED 3 x(xgU A W y ( y ~ P 4 Vz(zEx+zEy))) A Wx ( 3 U x E U +3y x E y E C ) ) . -+
-+
--f
+
Ax GC i s composed o f two p a r t s . The f i r s t , which w i l l be c a l l e d AxC, says t h a t f o r any c l a s s A o f d i s j o i n t nonempty sets, t h e r e i s a c l a s s B t h a t c o n t a i n s e x a c t l y one element o f each s e t i n A. (Thus, AxC i s 3 z Z E XA 1 3 z ( z E x A z E y ) ) WXW g ( x , q E A A x # q 3 B W x(xEA 3! q ( y E 8 A EX)).) The second, a s s e r t s t h a t o u r u n i v e r s e V i s t h e u n i o n of a c l a s s C of s e t s w e l l - o r d e r e d by i n c l u s i o n . T h i s axiom w i l l n o t be used u n t i l P a r t 4, where a c l a r i f i c a t i o n o f i t s meaning w i l l be given. +
-+
-+
Many doubts about t h e t r u t h o f t h i s axiom have been expressed. T h i s i s due, on t h e one hand, t o t h e f a c t t h a t Ax C a s s e r t s t h e e x i s t e n c e o f a c l a s s , which i s n o t unique, w i t h o u t d e f i n i n g i t , and, on t h e o t h e r hand, t o some o f i t s consequences which a r e strange.
9
A X I O M A T I C SET THEORY
However, Ax C i s i n d i s p e n s a b l e f o r numerous p r o o f s i n many d i f f e r e n t mathematical d i s c i p l i n e s , and i t has been shown c o n s i s t e n t w i t h t h e o t h e r axioms ( i f these a r e themselves c o n s i s t e n t ) . The t h e o r y i n c l u d i n g Ax GC besides those i n t h e p r e v i o u s s e c t i o n w i l l be denoted B C .
1.2.2.2
A X I O M S OF
F O U N D A T I O N S OR R E G U L A R I T Y
8
T h i s axiom excludes classes A f o r which t h e r e i s an i n f i n i t e sequence such t h a t E X E X E X ~ EA . *.. 2 1
...
xo, x l , ..., x n
I n o u r i n f o r m a l model, we s t a r t from an i n i t i a l c o l l e c t i o n o f U r e l e ment ( i n o u r case empty). A l l c l a s s e s a r e formed from t h i s c o l l e c t i o n . If a c l a s s A e x i s t e d w i t h t h e above mentioned p r o p e r t y , then A would n o t be o b t a i n e d from Urelemente. I n o r d e r t o c l a r i f y t h i s m a t t e r , l e t us c a l l t h e k e r n e l o f a c l a s s B , t h e c o l l e c t i o n o f Urelemente which e i t h e r b e l o n g t o B y o r t h e elements o f B, o r t o elements o f elements o f B y e t c . The Axiom of R e g u l a r i t y c o u l d be k e t i m L f r . Combining t h i s p r i n c i p l e w i t h t h e paraphrased: "EvefLy &!cud buu non e x i s t e n c e o f Urelemente ( o b t a i n e d f r o m Ax E x t ) we a r r i v e t o : "EvefLq c&& A b u X t 6hum Rhe empty some.
The f o r m u l a t i o n o f t h i s s o r t o f axiom i n d: would be e x t r e m e l y cumberTherefore, we adopt: P Ax Reg.
3 x xEA
+
3 x(xEA A V Y ( q E x
+
q P A)).
I t i s easy t o show t h a t Ax Reg excludes i n f i n i t e €-descending sequences o f t h e form x E X E x O , by c o n s i d e r i n g t h e class A = {xo, x1 , 2 1 x2, 1.
...
...
We s h a l l n o t o f f i c i a l l y adopt t h i s axiom i n o u r t h e o r i e s . However, we s h a l l mention t h e o r i e s i n which t h i s axiom i s v a l i d . I f T i s any t h e o r y T R w i l l be T w i t h Ax Reg added.
1.2
Axioms
The axiom who f o r m u l a t e d p l e s discussed can be deduced
1.2.1
system t o be i n t r o d u c e d i n t h i s s e c t i o n i s due t o P. Bernays i t i n 1961 (see Bernays 1976) i n s p i r e d by r e f l e c t i o n p r i n c i b y Montague and LPvy (LPvy 1960). A l l theorems i n t h i s book from i t .
GENERAL AXIOMS,
We f i r s t d i s c u s s a s e t o f axioms which can be j u s t i f i e d by t h e conside r a t i o n s s e t f o r t h i n t h e p r e v i o u s s e c t i o n . A l a r g e p a r t o f s e t t h e o r y can be developed from them. 1.2.1.1
I M P R E D I C A T I V E A X I O M OF C L A S S S P E C I F I C A T I O N ,
T h i s axiom has been proposed by s e v e r a l people. The f i r s t mentions of i t seems t o be by Q u i n e ( Q u i n e 19511, Morse (Morse 19651, K e l l e y ( K e l l e y 1955, Appendix) and Mostowski (Mostowski 1950). I t a s s e r t s t h e e x i s t e n c e o f a c l a s s t h a t c o n t a i n s a l l s e t s s a t i s f y i n g a g i v e n c o n d i t i o n we m i g h t f o r m u l a t e t h i s axiom by:
Fa& w a y p m p e n t y P thane A a d a b s A calzclidfincj
t h a t P (XI,
04
th.e
hcth
x nuch
However, t h i s f o r m u l a t i o n where P i s a v a r i a b l e r e f e r r i n g t o p r o p e r t i e s . i s i n second-order l o g i c , because i t c o n t a i n s q u a n t i f i c a t i o n over propert i e s . Thus, we have t o r e p l a c e t h i s axiom b y f i r s t - o r d e r axioms.
A n a t u r a l method f o r t h i s purpose, i s t o r e p l a c e p r o p e r t i e s by f i r s t o r d e r formulas. T h i s procedure was proposed by Skolem and Fraenckel
.
I n o r d e r t o do t h i s , we have t o d e f i n e c l e a r l y t h e l o g i c a l f i r s t - o r d e r language. A l l axioms w i l l be formulated i n t h e p r i m i t i v e language e which P w i l l be c o n s t i t u t e d as f o l l o w s :
A ) V a r i a b l e s . V a r i a b l e s a r e lower-case and c a p i t a l s c r i p t l e t t e r s w i t h o r w i t h o u t s u b s c r i p t . For i n s t a n c e A , x 0,
....
B) Constants.
-
( i ) L o g i c a l constants: 1 ( t h e n e g a t i o n symbol), + ( i m p l i c a t i o n symb o l ) , V ( d i s j u n c t i o n symbol), A ( c o n j u n c t i o n symbol), (equivalence symbol), V (universal quantifier), 3 (existential quantifier), 3 ! (the q u a n t i f i e r ' t h e r e e x i s t s e x a c t l y one'), = ( t h e i d e n t i t y symbol), and (,) parentheses).
5
6
ROLAND0 C H U A Q U I
(ii) N o n l o g i c a l constants. The b i n a r y pred c a t e An e x p / r e h i o n o f
eP
E
( t h e membership
r e l a t i o n symbo ).
i s a f i n i t e sequence o f symbols.
Among t h e ex-
p r e s s i ons, we d i s t i n g u i s h t h e formulas: (1) I f x , y a r e v a r i a b l e s , t h e n
x
=
q and x € y a r e formulas.
( 2 ) I f J, and 9 a r e formulas, and x i s a v a r i a b l e , then 1 9 , ( 9 + $ ) ( @ " G), f@-* $1, (9 s)@'+ Wx 9, 3 x 9 , and I ! x 9 a r e formulas. ( 3 ) A l l formulas o f L
o f (1) and ( 2 ) .
P
,
a r e o b t a i n e d by a f i n i t e number o f a p p l i c a t i o n
Greek l e t t e r s w i l l denote expressions.
9 , J,, 0 w i l l be formulas.
Bound and f r e e v a r i a b l e s a r e d e f i n e d as usual. Sentences a r e formul a s w i t h o u t f r e e v a r i a b l e s . We say t h a t t h e f o r m u l a J, i s a ( u n i v e r s a l ) W xn 9. c l o s u r e o f t h e formula 9 , i f J, i s a sentence o f t h e form W xo
...
A l l axioms, d e f i n i t i o n s , and theorems o f o u r t h e o r i e s a r e sentences. When a s s e r t i n g such sentences t h e i n i t i a l s t r i n g o f u n i v e r s a l q u a n t i f i e r s w i l l be g e n e r a l l y o m i t t e d . I n o t h e r words, when a s s e r t i n g a c l o s u r e o f 9 , we s h a l l u s u a l l y w r i t e 9 . I n w r i t i n g formulas, some parentheses w i l l be g e n e r a l l y o m i t t e d . Those o m i t t e d should be r e s t o r e d as f o l l o w s : F i r s t proceed from l e f t t o r i g h t and when a q u a n t i f i e r i s reached, we a s s i g n t o i t t h e s m a l l e s t p o s s i b l e scope. We r e p e a t t h i s process f i r s t f o r 1 , t h e n f o r A and V , t h e n f o r +, G X [ y ] is t h e formula o b t a i n e d from 9 by s u b s t i t u t and, f i n a l l y f o r f-r i n g a l l f r e e occurrences o f x by Y. L a t e r we s h a l l i n t r o d u c e e x t e n s i o n s o f L by adding new n o n l o g i c a l P symbol s.
.
Our f i r s t axiom, o r r a t h e r axiom-schema i s t h e f o l l o w i n g : AxClass (Schema), Fon m y 6ohmuf.u 9 0 6 I: which d o u n o t contain A P @ee, any d o ~ w r e06 ,the 6oUowing dotunLLea A an axiom: 3 A tlx(xEA-@~A 3 U x E U ) .
Note t h a t t h i s schema (as a l l schemata) r e p r e s e n t s an i n f i n i t e number @ may c o n t a i n any o t h e r f r e e vao f axioms, one f o r each f o r m u l a @ o f L P' r i a b l e s besides x ; t h e o n l y f r e e v a r i a b l e excluded i s A. '3 U x E U ' expresses ' x i s a s e t ' .
1.2.1.2
AXIOM OF EXTENSIONALITY
8
T h i s axiom s t a t e s t h e e s s e n t i a l p r o p e r t y o f c l a s s e s as extensions o f p r o p e r t i e s , namely, t h a t c l a s s e s a r e determined b y t h e i r elements.
AXIOMATIC S E T T H E O R Y
7
The converse implication i s a l s o t r u e by v i r t u e of the laws of logic.
T h i s axiom excludes Urelemente, s i n c e i t implies t h a t t h e r e i s j u s t one o b j e c t with no elements, namely, t h e empty c l a s s .
AXIOM OF SUBSETS,
1.2.1.3
This axiom expresses t h e f a c t t h a t our universe i s closed under ' s e t of . . . I where instead of we p u t any element of t h e universe, i.e. any s e t . T h u s , a subclass of a s e t should be a s e t .
...
3 U bell A Wx(xEa+ xEb) + 3 U aeU.
Ax Sub. 1.2.1.4
AXIOM OF REFLECTION,
This fourth axiom enbodies t h e p r i n c i p l e t h a t i f we have a universe Y already given, then we can take V as a s e t of Urelemente in order t o form another universe V . T h u s , V will be a s e t in t h i s new universe V ' and any c l a s s A (subclass of V ) will a l s o be a s e t i n V ' .
...',
We assumed V t o be closed under 'element of i.e. any element o f an element of V i s an element of V . Thus, I/ a s a s e t i n V ' should have t h i s same property. Before introducing t h e Axiom of Reflection, we need some notation. For each @ of I: t h a t does not contain U, will be the formula obtained by P r e l a t i v i z i n g each bound v a r i a b l e X t o the formula V q ( y E X --* y E U ) (i.e. t o t h e formula t h a t says t h a t X i s a subclass of U). This formula will be abbreviated by X 5 U. More p r e c i s e l y , @u i s defined .by recursion a s follows: ( i ) If @ i s x E y ( i i ) If @ i s 0 vel y.
or
-,dJ
x = y
or 10
( i i i ) I f @ i s WX 0
, then
(iv) If @ i s 3 X 0
, then
@'
, then
, then
@
U
@
U
is @ itself.
i s O U + dJu or 1 OU, U
is
W X(X c -U
is
3X(X c - U A 0').
+
respecti-
0 ).
All other logical connectives can be defined in terms of these, so we do not need t o include them i n the d e f i n i t i o n . Notice t h a t i f @ i s 3 ! X 0 , then @ U i s equivalent t o 3 ! X(X c - U A 0'). The axiom schema of r e f l e c t i o n i s , then, AxRef. (Schema). b and containn at W A(@ YEA
+
m0b.t
3 u( 3 U
A y E u ) +@:[
bl
Fox each am& @ 06 I: t h a t doen n o t c o n t a i n u o h P A Q~e-e,t h e 6o&eowing 0 an axiom: uEU
1.
A W
X W q ( x ~ q ~ ux e u ) -+
A W b( W q(yEb
++
8
R O L A N D 0 CHUAOUI
T h i s axiom says t h a t i f A i s a c l a s s t h a t s a t i s f i e s t h e p r o p e r t y dJ, then t h e r e i s a s e t u, which i s t r a n s i t i v e (i.e. X E U i m p l i e s x C - u), such t h a t t h e common p a r t o f A and u, i.e. A n u , s a t i s f i e s dJu. The j u s t i f i c a t i o n o f Ax Ref based on t h e i n f o r m a l n o t i o n s runs as f o l Suppose we have a c l a s s A p o s e s i i n g a p r o p e r t y dJ, i.e. such t h a t I ; t h i s c l a s s i s a subclass o f our u n i v e r s e Y . I t i s c l e a r t h a t @[A] V i s e q u i v a l e n t t o dJ [ A ] , because a l l c l a s s e s a r e subclasses o f V . We now t a k e V and form another u n i v e r s e V ' such t h a t V E V ' . Then, i n t h i s new universe, V i s a s e t u. Since V i s c l o s e d under 'element o f . . . I , u satisfies:
lows. 9 [A
x W y(x€q€u Also, A c - u. i s t h e same as A . V as dJ [ A ] .
B.
Thus, i f b s a t i s f i e s
v
-+
XEU)
y(qEb
Therefore, s i n c e V i s u , dJ:[bl
.
-
Y E A A ~ E u ) ,t h e n i t
expresses t h e same f a c t
The t h e o r y o b t a i n e d w i t h t h e axioms o f t h i s s e c t i o n w i l l be denotedby 1.2.2
L I M I T I N G AXIOMS,
The f o l l o w i n g axioms exciude c e r t a i n c l a s s e s and make t h e development o f t h e t h e o r y simpler. However, t h e y a r e n o t as w e l l j u s t i f i e d as t h e prev i o u s ones. These new axioms have been e x t e n s i v e l y by s t u d i e d metamathem a t i c a l l y , e s p e c i a l l y w i t h r e g a r d t o q u e s t i o n s about t h e i r c o n s i s t e n c y and independence. Thus, i t i s convenient t o deal s e p a r a t e l y w i t h them. 1.2.2.1
AXIOM OF GLOBAL CHOICE,
T h i s axiom excludes s e t s and c l a s s e s t h a t cannot be w e l l ordered (see 2.2.3.21 f o r a d e f i n i t i o n o f w e l l o r d e r i n g s ) . Ax GC.
v x tj y ( x , q E A A x # q 3 z Z E XA 1 3 z ( z E x A z E y ) ) 3 8 V x ( x E A -, 3! ~ ( q E A x y E B ) ) A 3 C( W U ( W x ( x E U x E C ) A 3 x XED 3 x(xgU A W y ( y ~ P 4 Vz(zEx+zEy))) A Wx ( 3 U x E U +3y x E y E C ) ) . -+
-+
--f
+
Ax GC i s composed o f two p a r t s . The f i r s t , which w i l l be c a l l e d AxC, says t h a t f o r any c l a s s A o f d i s j o i n t nonempty sets, t h e r e i s a c l a s s B t h a t c o n t a i n s e x a c t l y one element o f each s e t i n A. (Thus, AxC i s 3 z Z E XA 1 3 z ( z E x A z E y ) ) WXW g ( x , q E A A x # q 3 B W x(xEA 3! q ( y E 8 A EX)).) The second, a s s e r t s t h a t o u r u n i v e r s e V i s t h e u n i o n of a c l a s s C of s e t s w e l l - o r d e r e d by i n c l u s i o n . T h i s axiom w i l l n o t be used u n t i l P a r t 4, where a c l a r i f i c a t i o n o f i t s meaning w i l l be given. +
-+
-+
Many doubts about t h e t r u t h o f t h i s axiom have been expressed. T h i s i s due, on t h e one hand, t o t h e f a c t t h a t Ax C a s s e r t s t h e e x i s t e n c e o f a c l a s s , which i s n o t unique, w i t h o u t d e f i n i n g i t , and, on t h e o t h e r hand, t o some o f i t s consequences which a r e strange.
9
A X I O M A T I C SET THEORY
However, Ax C i s i n d i s p e n s a b l e f o r numerous p r o o f s i n many d i f f e r e n t mathematical d i s c i p l i n e s , and i t has been shown c o n s i s t e n t w i t h t h e o t h e r axioms ( i f these a r e themselves c o n s i s t e n t ) . The t h e o r y i n c l u d i n g Ax GC besides those i n t h e p r e v i o u s s e c t i o n w i l l be denoted B C .
1.2.2.2
A X I O M S OF
F O U N D A T I O N S OR R E G U L A R I T Y
8
T h i s axiom excludes classes A f o r which t h e r e i s an i n f i n i t e sequence such t h a t E X E X E X ~ EA . *.. 2 1
...
xo, x l , ..., x n
I n o u r i n f o r m a l model, we s t a r t from an i n i t i a l c o l l e c t i o n o f U r e l e ment ( i n o u r case empty). A l l c l a s s e s a r e formed from t h i s c o l l e c t i o n . If a c l a s s A e x i s t e d w i t h t h e above mentioned p r o p e r t y , then A would n o t be o b t a i n e d from Urelemente. I n o r d e r t o c l a r i f y t h i s m a t t e r , l e t us c a l l t h e k e r n e l o f a c l a s s B , t h e c o l l e c t i o n o f Urelemente which e i t h e r b e l o n g t o B y o r t h e elements o f B, o r t o elements o f elements o f B y e t c . The Axiom of R e g u l a r i t y c o u l d be k e t i m L f r . Combining t h i s p r i n c i p l e w i t h t h e paraphrased: "EvefLy &!cud buu non e x i s t e n c e o f Urelemente ( o b t a i n e d f r o m Ax E x t ) we a r r i v e t o : "EvefLq c&& A b u X t 6hum Rhe empty some.
The f o r m u l a t i o n o f t h i s s o r t o f axiom i n d: would be e x t r e m e l y cumberTherefore, we adopt: P Ax Reg.
3 x xEA
+
3 x(xEA A V Y ( q E x
+
q P A)).
I t i s easy t o show t h a t Ax Reg excludes i n f i n i t e €-descending sequences o f t h e form x E X E x O , by c o n s i d e r i n g t h e class A = {xo, x1 , 2 1 x2, 1.
...
...
We s h a l l n o t o f f i c i a l l y adopt t h i s axiom i n o u r t h e o r i e s . However, we s h a l l mention t h e o r i e s i n which t h i s axiom i s v a l i d . I f T i s any t h e o r y T R w i l l be T w i t h Ax Reg added.