By J. Barwise

Booklet by way of Barwise, J.

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**Example text**

I. e. B is the set of sets in clpse(NuAl)) and note that the set of urelements in clpse(NuAl) is just N. 7. Since NuA 1 is extensional, f establishes an isomorphism between m 1 and a structure m=

11 on b... (ii) If XEb.. 11 on b... (iii) The ~1 relations of b.. are closed under /\, v, 3XEa, 't/XEa, 3x. 11 as a subset of b... Part (ii) follows from (i). , to a ~1 formula and the ~ formulas are closed under the operations mentioned. 11 Exercise. VI' F(x,y) = G(x,y, {F(x,z)lzETC(y)}) recursion. ) by ~ ~1 2. VI is hereditarily finite if TC(a) is finite. VI' It can also be defined by: HFM(O) =0; HFM(n + 1)=set of all finite subsets of (MuHFM(n»; lHFM=Un

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.