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

  • Robert S. Lubarsky (a1)


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?”


Corresponding author

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


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[C]Chong, C. T., Maximal sets and fragments of Peano arithmetic (to appear).
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[GS]Groszekand, M.Slaman, T. A., Foundations of the priority method. I: Finite and infinite injury (to appear).
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[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.
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[SW]Slaman, T. A. and Woodin, W. H., Σ1 collection and the finite injury priority method (to appear).


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