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Appendix

Appendix

APPENDIX 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 de...

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APPENDIX

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

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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

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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.