New PDF release: Admissible Sets and Structures: An Approach to Definability

By J. Barwise

ISBN-10: 3662110350

ISBN-13: 9783662110355

ISBN-10: 3662110377

ISBN-13: 9783662110379

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Example text

1 C. 5 Theorem. lJI. (See Fig. ) Proof. DI. DI. Let us check Ao Collection leaving the easier axioms as exercises to help the student master absoluteness arguments. Suppose a, bEAo and suppose Ao satisfies 'VxEa 3y

1 Theorem (Existence of Transitive Closure). We can introduce a ~ function symbol TC into KPU so that the following becomes a theorem of KPU: For every x, TC(x) is a transitive set such that x ~ TC(x); and for any other transitive set a, if x~a, then TC(x)~a. 1, in the form of Proof by Induction over E. If one takes the contrapositive of foundation one gets the following scheme. For every formula cp the following is a theorem of KPU: V'x(V'YEX cp(y) --+ cp(x))--+ V'xcp(x). Thus in proving V'x cp(x), we pick an arbitrary x and prove cp(x) using, in the proof, cp(y) for any yEX.

Given aEA the expressions TC(a), P(a) each have two possible interpretations. =(9Jl; A,E). DII= P(a)=cI. lJII=TC(a)=b o; and so bo = bl . Thus bo is the real transitive closure of a, so that bo=n{blb transitive, asb}. For P, however, this fails. Since x sy is Ao we get Co S CI but that's all. Typically, Co will be a proper subset of the real power set CI of a. 4 Defmitions. A pure set in \YM is a set a with empty support; i. , one with TC(a)nM =0. ) A pure admissible set is an admissible set which is a model of KP; i.

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Admissible Sets and Structures: An Approach to Definability Theory by J. Barwise


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