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§1. The Well Ordering Theorem
§l. THE WELL ORDERING THEOREM
CWO 1-4 are class forms of WO 1, CWO 5 is a class form
of WO 2, CWO 6 ( m ) - 8 are class forms of WO 4 ( m ) - 6 and CWO 9 is a class form of MC 2.
There is a function F(x) well orders x.
such that for each set
F
CWO 1:
There is a relation R ordered by R.
x,
such that every set is well
CWO 2:
Every class can be well ordered. universe, V, can be well ordered.)
CWO 3:
(Equivalently, the
CWO 4 :
There is a relation R which well orders V in such a way that each proper initial R-segment of V is a set. CWO 5: V
Each proper class is equipollent to
On.
(Equivalently,
On.) Let
m E
w
such that
m
2
1.
CWO 6 ( m ) :
on
On
CWO 7:
For each class X there is a function F defined U F(a) = X. such that for each a E On, F ( a ) < m and aeon There is an m E w {O} such that CWO 6 ( m ) .
-
-
For each class X there exists an m E w {O} and a function F defined on On such that for each a E On, F(a) 4 m and U F(a) = X . aEOn CWO 9: Every class X is the union of a well ordered class of finite sets. CWO 8:
187
PART 11, CLASS FORMS
188
I n e a c h of CWO 6 ( m ) - 9 , w e o b t a i n an e q u i v a l e n t s t a t e m e n t by r e p l a c i n g " X " by " V " . Thus, e a c h i m p l i e s t h a t t h e u n i v e r s e i s a w e l l o r d e r e d union of f i n i t e sets. The n e x t form, CWO 1 0 , i s a c l a s s form of P R O J g i v e n i n s e c t i o n 5.
I t was g i v e n a s a n axiom of s e t t h e o r y i n von
Neumann [1925
,
Axiom I V 21 a s a r e p l a c e m e n t f o r t h e w e l l
o r d e r i n g theorem and i n von Neumann [1928] it was proved t h a t CWO 1 0
+
CWO 1 0 :
CWO 3.
Every p r o p e r c l a s s c a n be mapped o n t o
every proper c l a s s
X, V
<*
The f o l l o w i n g s t a t e m e n t
V.
(For
X.)
i s a c l a s s form o f t h e
CT
t r i c h o t o m y , b u t it i s e a s i l y s e e n t o b e e q u i v a l e n t t o a form of t h e w e l l o r d e r i n g theorem.
CT:
Each p a i r of p r o p e r c l a s s e s i s
equipollent.
I t i s n o t known whether t h e s t a t e m e n t :
i s comparable.
Each p a i r o f p r o p e r c l a s s e s
i m p l i e s a c l a s s form of t h e axiom of c h o i c e .
Felgner
&
Flannagan [1978] have shown t h a t t h e s t a t e m e n t : On
c a n be i n j e c t e d i n t o e v e r y p r o p e r c l a s s .
d o e s n o t imply t h e s e t form o f t h e axiom o f c h o i c e i n NBG. CWO 11, CWO 1 2 , and CPW a r e c l a s s forms o f PW, CLW i s a
c l a s s form of LW, and CCW i s a c l a s s form o f WO 8. CWO 11:
For e a c h p r o p e r c l a s s
CWO 1 2 :
F(0n)
CPW:
X,
p(X) <
X.
c a n be w e l l o r d e r e d .
The power c l a s s o f a w e l l o r d e r e d c l a s s c a n be w e l l
ordered. CLW:
Every l i n e a r l y o r d e r e d c l a s s can b e w e l l o r d e r e d .
CCW: Every c l a s s on which t h e r e is a c h o i c e f u n c t i o n w e l l ordered.
The f o l l o w i n g i m p l i c a t i o n s a r e immediate.
can be
THE WELL O R D E R I N G THEOREM
51. CT
-
-
CWO 5
CWO 5
CWO 6 ( m ) CWO 6 ( m )
-
CWO 3 + CWO 2
-+
4
CWO
CWO 6 ( n )
-+
CWO 7 + CWO 8 -+
if
-
CWO 1;
m s n;
-+
CWO 5 + CWO 11
+
CWO 6 ( 1 )
CWO 10;
The p r o o f t h a t
CWO 8
-+
i s s i m i l a r t o t h e proof
CWO 5
I n f a c t , t h e proof h e r e i s a l i t t l e e a s i e r
v
and w e a u t o m a t i c a l l y have +
CPW,
CCW;
CWO 9 ;
+
b e c a u s e it i s s u f f i c i e n t t o t a k e t h e CLW
-
CPW -+ CWO 12.
o f Theoxem 1.1. "V"
CWO 3 CWO 5;
189
and
CWO 12
+
-+
"X"
v 5 V.
o f CWO 8 t o be The p r o o f s t h a t
CWO 4 a r e s i m i l a r t o t h e p r o o f s f o r
( S e e Theorems 5 . 5 and 5 . 7 and
t h e c o r r e s p o n d i n g set forms. n o t i c e t h a t CWO 1 2
x
The p r o o f t h a t CWO 1 2
PW.
+
CWO 4 i s
T o complete t h e proof o f e q u i v a l e n c e s w e g i v e t h e
i n NBG.)
f o l l o w i n g two t h e o r e m s . CWO 1
THEOREM 1 . 1 C :
PROOF:
For each
o r d i n a l number
a
u
+
CWO 1 2 .
5 On
let
such t h a t f o r a l l
be t h e smallest B E u, B < a. F o r e a c h
p(u)
xa = {u _c On: p ( u ) = a}. Then s i n c e x 5 ?(a), xa i s a s e t . L e t F b e t h e f u n c t i o n g u a r a n t e e d by CWO 1 s u c h
a, l e t
t h a t f o r each
w e l l order
a E On, F(x,) w e l l o r d e r s xa. Then w e c a n by a r e l a t i o n R a s f o l l o w s : f o r u , v 5 On,
?(On)
THEOREM 1 . 2 C :
PROOF: by
R.
CWO 9
Suppose
+
i s a c l a s s which i s l i n e a r l y o r d e r e d
X
i s a w e l l o r d e r e d u n i o n of A l i n e a r o r d e r i n g on a f i n i t e s e t i s a w e l l
CWO 9 i m p l i e s t h a t
f i n i t e sets. ordering.
CLW.
Thus, u s i n g
In Figure 1.1C,
R
X
we can w e l l o r d e r
X.
w e summarize t h e r e s u l t s o f t h i s s e c t i o n .
A l l i m p l i c a t i o n s were p r o v e d i n NBG'.
All t h e f o r m s i n t h e s e t @ are e q u i v a l e n t i n NBGO, where
@
= {CT, CWO 4, CWO 5, CWO 6 ( m ) , CWO 7, CWO 8 , CWO 101.
PART 11, CLASS FORMS
190
y
+ CWO 1
qcw y
@
CWO 2
ccw
3\
(3)
(3)
CLW
f
Y
CPW 4 CWO 12
11
FIGURE 1.1C
@ in NBG; thus all the We have shown that CWO 12 statements in Figure 1.1C are equivalent in NBG. (It was shown by Howard, Rubin & Rubin I19781 that the implications (1) and ( 2 ) are not reversible in NBGO, but a strongly inaccessible cardinal was used to prove ( 2 ) . Felgner & Jech [1973] proved that the implications ( 3 ) are not reversible in NBGO. The proofs in Felgner & Jech were for the corresponding set forms, but it is easy to see that they hold for the class forms also. It follows from model (e) in Felgner & Jech I19731 that ( 4 ) is not reversible in NBGO). -+
To conclude this section we shall show that CWO 2 along with DC, the Principle of Dependent Choices (see Part I, Section 2 ) implies CWO 3 . Clearly, CWO 3 -P DC. THEOREM 1.3C:
CWO 2
A
DC
-+
CWO 3 .
PROOF: Let W be an irreflexive relation which well orders each set. Suppose W does not well order V. Then there is a non-empty class A such that A has no W-first element. Define a relation R as follows: R Since
A
=
{(x,y)
E A x A: (y,x) E W}.
has no W-first element, $(R) = A. Therefore, DC implies that there is a function f such that for each n E w, (f(n),f(n+l)) E R. But, by the This definition of R, (f (n+l),f (n)) E W for each n E w. implies the contradiction that the set {f(n): n E w } has no W-first element, q.e.d.
g(R) 5 a(R) so
CWO 1-4 are class forms of WO 1, CWO 5 is a class form
of WO 2, CWO 6 ( m ) - 8 are class forms of WO 4 ( m ) - 6 and CWO 9 is a class form of MC 2.
There is a function F(x) well orders x.
such that for each set
F
CWO 1:
There is a relation R ordered by R.
x,
such that every set is well
CWO 2:
Every class can be well ordered. universe, V, can be well ordered.)
CWO 3:
(Equivalently, the
CWO 4 :
There is a relation R which well orders V in such a way that each proper initial R-segment of V is a set. CWO 5: V
Each proper class is equipollent to
On.
(Equivalently,
On.) Let
m E
w
such that
m
2
1.
CWO 6 ( m ) :
on
On
CWO 7:
For each class X there is a function F defined U F(a) = X. such that for each a E On, F ( a ) < m and aeon There is an m E w {O} such that CWO 6 ( m ) .
-
-
For each class X there exists an m E w {O} and a function F defined on On such that for each a E On, F(a) 4 m and U F(a) = X . aEOn CWO 9: Every class X is the union of a well ordered class of finite sets. CWO 8:
187
PART 11, CLASS FORMS
188
I n e a c h of CWO 6 ( m ) - 9 , w e o b t a i n an e q u i v a l e n t s t a t e m e n t by r e p l a c i n g " X " by " V " . Thus, e a c h i m p l i e s t h a t t h e u n i v e r s e i s a w e l l o r d e r e d union of f i n i t e sets. The n e x t form, CWO 1 0 , i s a c l a s s form of P R O J g i v e n i n s e c t i o n 5.
I t was g i v e n a s a n axiom of s e t t h e o r y i n von
Neumann [1925
,
Axiom I V 21 a s a r e p l a c e m e n t f o r t h e w e l l
o r d e r i n g theorem and i n von Neumann [1928] it was proved t h a t CWO 1 0
+
CWO 1 0 :
CWO 3.
Every p r o p e r c l a s s c a n be mapped o n t o
every proper c l a s s
X, V
<*
The f o l l o w i n g s t a t e m e n t
V.
(For
X.)
i s a c l a s s form o f t h e
CT
t r i c h o t o m y , b u t it i s e a s i l y s e e n t o b e e q u i v a l e n t t o a form of t h e w e l l o r d e r i n g theorem.
CT:
Each p a i r of p r o p e r c l a s s e s i s
equipollent.
I t i s n o t known whether t h e s t a t e m e n t :
i s comparable.
Each p a i r o f p r o p e r c l a s s e s
i m p l i e s a c l a s s form of t h e axiom of c h o i c e .
Felgner
&
Flannagan [1978] have shown t h a t t h e s t a t e m e n t : On
c a n be i n j e c t e d i n t o e v e r y p r o p e r c l a s s .
d o e s n o t imply t h e s e t form o f t h e axiom o f c h o i c e i n NBG. CWO 11, CWO 1 2 , and CPW a r e c l a s s forms o f PW, CLW i s a
c l a s s form of LW, and CCW i s a c l a s s form o f WO 8. CWO 11:
For e a c h p r o p e r c l a s s
CWO 1 2 :
F(0n)
CPW:
X,
p(X) <
X.
c a n be w e l l o r d e r e d .
The power c l a s s o f a w e l l o r d e r e d c l a s s c a n be w e l l
ordered. CLW:
Every l i n e a r l y o r d e r e d c l a s s can b e w e l l o r d e r e d .
CCW: Every c l a s s on which t h e r e is a c h o i c e f u n c t i o n w e l l ordered.
The f o l l o w i n g i m p l i c a t i o n s a r e immediate.
can be
THE WELL O R D E R I N G THEOREM
51. CT
-
-
CWO 5
CWO 5
CWO 6 ( m ) CWO 6 ( m )
-
CWO 3 + CWO 2
-+
4
CWO
CWO 6 ( n )
-+
CWO 7 + CWO 8 -+
if
-
CWO 1;
m s n;
-+
CWO 5 + CWO 11
+
CWO 6 ( 1 )
CWO 10;
The p r o o f t h a t
CWO 8
-+
i s s i m i l a r t o t h e proof
CWO 5
I n f a c t , t h e proof h e r e i s a l i t t l e e a s i e r
v
and w e a u t o m a t i c a l l y have +
CPW,
CCW;
CWO 9 ;
+
b e c a u s e it i s s u f f i c i e n t t o t a k e t h e CLW
-
CPW -+ CWO 12.
o f Theoxem 1.1. "V"
CWO 3 CWO 5;
189
and
CWO 12
+
-+
"X"
v 5 V.
o f CWO 8 t o be The p r o o f s t h a t
CWO 4 a r e s i m i l a r t o t h e p r o o f s f o r
( S e e Theorems 5 . 5 and 5 . 7 and
t h e c o r r e s p o n d i n g set forms. n o t i c e t h a t CWO 1 2
x
The p r o o f t h a t CWO 1 2
PW.
+
CWO 4 i s
T o complete t h e proof o f e q u i v a l e n c e s w e g i v e t h e
i n NBG.)
f o l l o w i n g two t h e o r e m s . CWO 1
THEOREM 1 . 1 C :
PROOF:
For each
o r d i n a l number
a
u
+
CWO 1 2 .
5 On
let
such t h a t f o r a l l
be t h e smallest B E u, B < a. F o r e a c h
p(u)
xa = {u _c On: p ( u ) = a}. Then s i n c e x 5 ?(a), xa i s a s e t . L e t F b e t h e f u n c t i o n g u a r a n t e e d by CWO 1 s u c h
a, l e t
t h a t f o r each
w e l l order
a E On, F(x,) w e l l o r d e r s xa. Then w e c a n by a r e l a t i o n R a s f o l l o w s : f o r u , v 5 On,
?(On)
THEOREM 1 . 2 C :
PROOF: by
R.
CWO 9
Suppose
+
i s a c l a s s which i s l i n e a r l y o r d e r e d
X
i s a w e l l o r d e r e d u n i o n of A l i n e a r o r d e r i n g on a f i n i t e s e t i s a w e l l
CWO 9 i m p l i e s t h a t
f i n i t e sets. ordering.
CLW.
Thus, u s i n g
In Figure 1.1C,
R
X
we can w e l l o r d e r
X.
w e summarize t h e r e s u l t s o f t h i s s e c t i o n .
A l l i m p l i c a t i o n s were p r o v e d i n NBG'.
All t h e f o r m s i n t h e s e t @ are e q u i v a l e n t i n NBGO, where
@
= {CT, CWO 4, CWO 5, CWO 6 ( m ) , CWO 7, CWO 8 , CWO 101.
PART 11, CLASS FORMS
190
y
+ CWO 1
qcw y
@
CWO 2
ccw
3\
(3)
(3)
CLW
f
Y
CPW 4 CWO 12
11
FIGURE 1.1C
@ in NBG; thus all the We have shown that CWO 12 statements in Figure 1.1C are equivalent in NBG. (It was shown by Howard, Rubin & Rubin I19781 that the implications (1) and ( 2 ) are not reversible in NBGO, but a strongly inaccessible cardinal was used to prove ( 2 ) . Felgner & Jech [1973] proved that the implications ( 3 ) are not reversible in NBGO. The proofs in Felgner & Jech were for the corresponding set forms, but it is easy to see that they hold for the class forms also. It follows from model (e) in Felgner & Jech I19731 that ( 4 ) is not reversible in NBGO). -+
To conclude this section we shall show that CWO 2 along with DC, the Principle of Dependent Choices (see Part I, Section 2 ) implies CWO 3 . Clearly, CWO 3 -P DC. THEOREM 1.3C:
CWO 2
A
DC
-+
CWO 3 .
PROOF: Let W be an irreflexive relation which well orders each set. Suppose W does not well order V. Then there is a non-empty class A such that A has no W-first element. Define a relation R as follows: R Since
A
=
{(x,y)
E A x A: (y,x) E W}.
has no W-first element, $(R) = A. Therefore, DC implies that there is a function f such that for each n E w, (f(n),f(n+l)) E R. But, by the This definition of R, (f (n+l),f (n)) E W for each n E w. implies the contradiction that the set {f(n): n E w } has no W-first element, q.e.d.
g(R) 5 a(R) so