By Robert J., Jr. Feugate, Steven M. McIntyre

ISBN-10: 0134988663

ISBN-13: 9780134988665

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But this follows from the fact that 10)= [l(u)I (u E y f B)] l and that for almost all u , u E B and I(u) = u. Now we turns to Theorem 8, and prove 2A = A+. First we suppose h 2 K (this case is due to Solovay). It will suffice to prove PS(A) C C,((h+), since this gives 2 A = IPS(h) I Q 1 C,((h+) I = A+ ' Let x C A. Choose A so that h U {x}C A and A < V . By the LowenheimSkolem theorem, choose B so that h U {x} C B < A < V and I B I Q A. Let 1 be the collapsing isomorphism of B onto D. Since h U {x} is transitive, I ( x ) = x ; s o x ED.

Using the transitivity of A , we get (7 €A ++ u < Ix(A) . It follows that LA = [C(U) Iu < IX(A)] = C*(IX(A)) The sentence V=L is called the axiom of constructibility. By the above, LL = L = V L ;so V=L is true in L. Lemma 6 . If A is a transitive model of ZFC containing all the ordinals, then L CA. R. SHOENFIELD Proof. L = L A C A . This lemma shows that L is the smallest model satisfying (L2). If there is a measurable cardinal, then V f L. Proof. Let K be the first measurable cardinal, and let the notation be as " i e .

If L - is the submodel of 311 of constructible sets, then L 3n I= 3 x8 l ( x ) by the Shoenfield-Lihy Lemma. Also note that L cM i= Q t V=L since Q is absolute for L. Then L p is a pointwise definable model of Q + V=L + 3x8 ,(x) since D(Lw ) = Lnc . Now since L p i= 3 x8 ,(x), L p k 3 x [ ITC(x) ((al A 8 ,(x)] by INFINITARY METHODS IN THE MODEL THEORY OF SET THEORY 57 Theorem 36 of Levy (or use the Shoenfield-Levy lemma again). Let (1, be a derivation in L , such that Let y be the least ordinal not in TC( (1,) so that L , k y < ol.

### Introduction to Vlsi Testing by Robert J., Jr. Feugate, Steven M. McIntyre

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