By J. Barwise

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13 Notes. The considerations involved in the choice of Ao Separation and Ao Collection are suggested by the informal notion of "predicative". Kripke, in fact, called his axioms for admissible sets PZF, for predicative ZF. As an explication of the intuitive idea of predicativity, however, KP has certain debatable features. See, for example, Feferman [1975] for a discussion and examples of set theories which are predicative in a more stringent sense. 38 I. Admissible Set Theory 9. Additional Axioms There are certain extensions of KPU which surface from time to time.

15. Prove that VM(w) is admissible iff M is finite. 16. Prove that H(l)={n1, ... ,nd, where 1=2n1 +···+2nk , n1 >···>nk , is a operation of 1. 17 Notes. 5. 7, will be quite important in Chapters IV and VI when dealing with structures without much coding machinery built into them. 6. The proof uses results from later chapters. lJI. lJI. lJI is semi-search computable. Let T be the diagram of 9Jl plus the axioms KPU coded up on M* by means of the pairing function and let S(x) iff "x codes a sentence provable from T'.

16. Prove that H(l)={n1, ... ,nd, where 1=2n1 +···+2nk , n1 >···>nk , is a operation of 1. 17 Notes. 5. 7, will be quite important in Chapters IV and VI when dealing with structures without much coding machinery built into them. 6. The proof uses results from later chapters. lJI. lJI. lJI is semi-search computable. Let T be the diagram of 9Jl plus the axioms KPU coded up on M* by means of the pairing function and let S(x) iff "x codes a sentence provable from T'. It is implicit in Chapter V (and explicit in Chapter VIII) that S is a complete ~1 prediciate.

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