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Nonrecursive combinatorial functions

  • Erik Ellentuck (a1)

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In this paper we show how nonrecursive combinatorial functions can be used to obtain another proof of the compactness Theorem 3.1 of [6]. Our method is closely related to the argument used to show that a countable ultraproduct is ℵ1-saturated. What we do is to diagonalize in the most obvious way. The main difficulty with this approach is that the resulting diagonal function need not be recursive. Just to give an idea of how bad nonrecursive combinatorial functions are, we mention that by using frames (cf. [5]) they do not extend to Λ, and that by using normal combinatorial operators (cf. [4]) they do extend to Λ, map Λ into Λ, but in general, composition of such functions does not commute with their extension. We take care of this problem by constructing a large class of isols with respect to which the diagonal function is well behaved. The advantage of our method is that it provides the investigator with a natural and intuitive way of constructing counterexamples in his own research area.

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[1]Dekker, J. C. E. and Myhill, J., Retraceable sets, Canadian Journal of Mathematics, vol. 10 (1958), pp. 357373.
[2]Ellentuck, E., Universal isols, Mathematische Zeitschrift, vol. 98 (1967), pp. 18.
[3]Ellentuck, E., Nonrecursive relations among the isols, Proceedings of the American Mathematical Society (to appear).
[4]Myhill, J., Recursive equivalence types and combinatorial functions, Bulletin of the American Mathematical Society, vol. 64 (1958), pp. 373376.
[5]Nerode, A., Extensions to isols, Annals of Mathematics, vol. 73 (1961), pp. 362403.
[6]Nerode, A., Diophantine correct nonstandard models in the isols, Annals of Mathematics, vol. 84 (1966), pp. 421432.

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