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We assume that M = C,”’(O) is a model of ZF; thus, in particular, M can be the minimal model. I f f is an increasing continuous uniformly definable function THEOREM. with domain On and range G On, then .for any ci in n ( M ) the ordinal w , ( M ) is a critical number o f f . PROOF. We first consider the case where ci = p+ 1. Let F be a formula with two free variables xo, x1which defines f (cf. V.l and IV.1). Hence for almost all sets A and arbitrary E, q in n(A)

f ( 8E A ,

(1)

[q =f(t)l= I - A m , 61. The correlation of the variables of F and the ordinals q, 6 is this: q is correlated with x, and 6 with xl. Thus the right-hand side of (2) should be written as k A F [ { ( O ,q), (1 , [ ) } I . The expression “for almost all” means that there are finitely many axioms K I, ... , K, of ZF such that whenever A is a transitive set in which these axioms are valid, then formulae (l), (2) hold for arbitrary 6,q < n(A). Let us assume that 6 < w,(M) and put s = o u 6 u (61.It is obvious that s is M-embeddable in o , ( M ) : (2)

ISI

Ghf los(WI-

Let H be the conjunction of the following formulae: the axioms K , , ... , K , , the formula Ord(x,) and the formula

_-I

G: (xl){Ord(x,) + (E!xo)[Ord(xo)A F ] ).

The last formula “says” that for each ordinal x, there is exactly one

xo such that F(xo, xl).

We can assume that the axiom of extensionality is among the axioms

256

APPENDIX

Ki.Since all the axioms of Z F are valid in M , we find from (1) and (2) that for all 5 < n ( M ) (3) (4)

(5)

t M G ,

-MKI/I

... A K,,

k OrW.

We now use theorem V11.7.3, in which we put A . = M and replace 5 by o p ( M )in ; view of the definition of M we also have A ; = M , where the meaning of A ; is explained in VII.7.3. Hence we obtain a set m in M such that

(6) ( 7)

Iml GlqdMIl,

c m,

rn is an H-elementary subset of M .

The contracting function g~ of m belongs to M (VII.6.5). Putting m* = Im(p, m) we obtain therefore a set m* E M such that (8) (9) (10) (1 1)

Im*l < u lq(M)I p(x) = x

for

9

x ES,

rp maps m isomorphically onto m*,

m* is transitive.

(10) and (1 1) follow from the properties of the contracting function (see 1.6). (8) results from (6) and the remark that q~ E M and g~ is oneto-one. Finally, (9) results from the theorem stating that the contracting function of m is equal to identity on each transitive subset of m (cf. 1.6.3). Put 5 = E in (4) and (5). By (7) and (6) we obtain 'v-,Ord[E] and from (3) and (7) we infer that ;-,,,G, and hence there is an element y e m such that +,,,F'[y, E] and k,Ordb]. Applying the isomorphism g~ and putting q = q(y), we obtain therefore

b- ,"+ F [ r El, Y

because g~ transforms 5 into itself. It follows that is an ordinal.

I- **Ord[ql

257

APPENDIX

From (2) we now obtain 17 =f(E). and hence 1171 \(& Iwp(M)I. Thus

If(0I < M

Since 17

E m*, we

have 17

c m*

los(M>I

and therefore

f(t)< "l9+1(M)= w a ( M ) . Thus we have shown that

[t < w,(M)l

+

y(cn< %(WJ,

and hence w,(M) is a critical number off. If tl is a limit number, then w,(M) = lim w,(M), and the theorem B

results from the previous case because a limit of critical numbers is itself a critical number. APPLICATIO xr. If we take in the theorem f(t) = d(E) and tl = w , we obtain

[B < %J(M)J

+

[W)< w,(M)I,

%(M) = d(w,(W). These two formulas were used in XIV.7, p. 243. In a similar way we can derive the formula d(o,(M)) = w , ( M ) ,which we used on p. 252.

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