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By J. Barwise

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Jl as a set and as an L*-structure, but it is usually clear which we have in mind. 1. (ii) is trivial since A must be closed under pair and union so that HFM(n)c:;A for all n, by induction on n. Jl is admissible. Jl we get extensionality and foundation for free. Jl(n) is also transitive. Jl(n+1) so we have Pair. Jl(n+ 1), and we have Union. Jl(n) since a subset of a finite set is finite, so we have full separation, hence ~o Separation. Jl(n), hence {x1, ... Jl(n+ 1). 2 Corollary. The smallest admissible set is IHF = {aE \Y Ia is a pure hereditarily finite set).

For example, let aE21m. ::\0 Collection. 6 that we have asserted separation and collection for absolute formulas, at least some of them. JI2 end 21m, the equation for b will remain true. Have we asserted separation and collection for all absolute formulas? Yes, but not explicitly. ::\0; separation for such q> is not an axiom of KPU. 8 Theorem. For any theory T of L*, if q>(x1, ... ,xn) is persistent relative to T then there is a ~ formula ljJ(Xl' ... , xn) such that TI--'v'Xl, ... , Xn [qJ(Xl, ...

E. same theorems) as adding all the following axioms, where cp is [\0: 3b \fxEa [3y cp(x,y) -+ 3YEb cp(x,y)]. 13 Notes. In a theory like ZF containing I:l Separation, Beta becomes a theorem and the collapsing lemma of § 7 is a consequence of it. In such theories Beta itself is often called The Collapsing Lemma. It is due to Mostowski [1949]' In KPU we must separate the two aspects since one is provable and the other is not. Beta is so named because Mostowski [1961] used the terminology "p-model" (with "(f' for bon ordre) for models where well-orderings were absolute.

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