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An introduction to γ-recursion theory (or what to do in KP – Foundation)

  • Robert S. Lubarsky (a1)

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The program of reverse mathematics has usually been to find which parts of set theory, often used as a base for other mathematics, are actually necessary for some particular mathematical theory. In recent years, Slaman, Groszek, et al, have given the approach a new twist. The priority arguments of recursion theory do not naturally or necessarily lead to a foundation involving any set theory; rather, Peano Arithmetic (PA) in the language of arithmetic suffices. From this point, the appropriate subsystems to consider are fragments of PA with limited induction. A theorem in this area would then have the form that certain induction axioms are independent of, necessary for, or even equivalent to a theorem about the Turing degrees. (See, for examples, [C], [GS], [M], [MS], and [SW].)

As go the integers so go the ordinals. One motivation of α-recursion theory (recursion on admissible ordinals) is to generalize classical recursion theory. Since induction in arithmetic is meant to capture the well-foundedness of ω, the corresponding axiom in set theory is foundation. So reverse mathematics, even in the context of a set theory (admissibility), can be changed by the influence of reverse recursion theory. We ask not which set existence axioms, but which foundation axioms, are necessary for the theorems of α-recursion theory.

When working in the theory KP – Foundation Schema (hereinafter called KP), one should really not call it α-recursion theory, which refers implicitly to the full set of axioms KP. Just as the name β-recursion theory refers to what would be α-recursion theory only it includes also inadmissible ordinals, we call the subject of study here γ-recursion theory. This answers a question by Sacks and S. Friedman, “What is γ-recursion theory?”

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Corresponding author

Department of Mathematics, Franklin and Marshall College, Lancaster, Pennsylvania 17604

References

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[B]Barwise, J., Admissible sets and structures, Springer-Verlag, Berlin, 1975.
[C]Chong, C. T., Maximal sets and fragments of Peano arithmetic (to appear).
[D]Devlin, K., Aspects of constructibility, Lecture Notes in Mathematics, vol. 354, Springer-Verlag, Berlin, 1973.
[Dr]Driscoll, G. C. Jr., Metarecursively enumerable sets and their metadegrees, this Journal, vol. 33 (1968), pp. 389411.
[F1]Friedman, Sy D., β-recursion theory, Transactions of the American Mathematical Society, vol. 255 (1979), pp. 173200.
[F2]Friedman, Sy D., Post's problem without admissibility, Advances in Mathematics, vol. 35 (1980), pp. 3049.
[GS]Groszekand, M.Slaman, T. A., Foundations of the priority method. I: Finite and infinite injury (to appear).
[L1]Lubarsky, R. S., Admissibility spectra and minimality, Annals of Pure and Applied Logic (to appear).
[L2]Lubarsky, R. S., Correction to [L3], this Journal, vol. 53 (1988), pp. 103104.
[L3]Lubarsky, R. S., Simple r.e. degree structures, this Journal, vol. 52 (1987), pp. 208213.
[M]Mytilinaios, M. E., Finite injury and Σ2 induction, this Journal, vol. 54 (1989), pp. 3849.
[MS]Mytilinaios, M. E. and Slaman, T. A., Σ2 collection and the infinite injury priority method, this Journal, vol. 53 (1988), pp. 212221.
[Sa]Sacks, G. E., Post's problem, admissible ordinals, and regularity, Transactions of the American Mathematical Society, vol. 124 (1966), pp. 123.
[Sh]Shore, R. A., Splitting an α-recursively enumerable set, Transactions of the American Mathematical Society, vol. 204 (1975), pp. 6578.
[SW]Slaman, T. A. and Woodin, W. H., Σ1 collection and the finite injury priority method (to appear).

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