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The positive properties of isolic integers

  • Erik Ellentuck (a1)

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In this paper we show (cf. Theorem 22) that in a language L* with equality, whose relation symbols denote arbitrary relations over ω* (=rational integers) and whose function symbols denote (= ∃∀ definable in the arithmetic hierarchy) functions over ω*, (i) a positive sentence is true in Λ* (= isolic integers) iff some Horn reduct is true in ω* with Skolem functions. We also show (cf Theorem 20) that (ii) a universally quantified sentence is true in Λ* iff some Horn reduct is true in Λ*.The latter result is nontrivial because our relations are arbitrary and our functions are In order to obtain (i) it was necessary to generalize the frame extensions of [7]. This is done in §2. Our extension procedure agrees with that of [7] for recursive relations (cf. Theorem 12), and is certainly more general for relations. What happens in the case is still open. In §3 we develop the basic properties of our extension so that in §4 we can prove a metatheorem (cf. Theorems 8 and 10) about Λ (=isols), in a language L with equality whose relation symbols denote arbitrary relations over ω (=nonnegative integers) and whose function symbols denote almost R↑ combinatorial functions. In Theorem 11 this is generalized to infinitary universal sentences. In §5 generic isols are introduced. These are used (cf. Theorems 16–19) to generalize and simplify the “fundamental lemma” of [8]. The basic induction is patterned after Lemma 4.1 of [8], but is stronger in that any sufficiently generic assignment attainable from a frame yields Skolem functions. Finally in §6 these results are applied to Λ*, yielding the titled result (i) of our paper. Immediately following Theorem 15 there is a discussion which attempts to justify the way we extend relations to Λ.

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[1]Cohen, P. J., Set theory and the continuum hypothesis, Benjamin, New York, 1966.
[2]Ellentuck, E., Universal isols, Mathematische Zeitscrift, vol. 98 (1967), pp. 18.
[3]Ellentuck, E., Extension methods in cardinal arithmetic, Transactions of the American Mathematical Society, vol. 149 (1970), pp. 307325.
[4]Ellentuck, E., Nonrecursive combinatorial functions, this Journal, vol. 37 (1972), pp. 9095.
[5]Ellentuck, E., Nonrecursive relations among the isols, to appear.
[6]Gold, E. M., Limiting recursion, this Journal, vol. 30 (1965), pp. 2848.
[7]Nerode, A., Extensions to isols, Annals of Mathematics, vol. 73 (1961), pp. 362403.
[8]Nerode, A., Extensions to isolic integers, Annals of Mathematics, vol. 75 (1962), pp. 419448.
[9]Nerode, A., Combinatorial series and recursive equivalence types, Fundamenta Mathematicae, vol. 58 (1966), pp. 113141.
[10]Nerode, A., Diophantine correct non-standard models in the isols, Annals of Mathematics, vol. 84 (1966), pp. 421432.

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