Friedberg Numbering in Fragments of Peano Arithmetic and α-Recursion Theory

2013 ◽  
Vol 78 (4) ◽  
pp. 1135-1163 ◽  
Author(s):  
Wei Li
Keyword(s):  
Recursion Theory ◽  
Peano Arithmetic ◽  
Fragments Of Peano Arithmetic

AbstractIn this paper, we investigate the existence of a Friedberg numbering in fragments of Peano Arithmetic and initial segments of Gödel's constructible hierarchy Lα, where α is Σ1 admissible. We prove that(1) Over P− + BΣ2, the existence of a Friedberg numbering is equivalent to IΣ2, and(2) For Lα, there is a Friedberg numbering if and only if the tame Σ2 projectum of α equals the Σ2 cofinality of α.

scholarly journals Maximal sets and fragments of Peano arithmetic

1989 ◽  
Vol 115 ◽  
pp. 165-183 ◽  
Author(s):  
C.T. Chong
Keyword(s):  
Recursion Theory ◽  
Peano Arithmetic ◽  
Base Theory ◽  
Abstract Principle ◽  
Priority Method ◽  
General Program ◽  
Fragments Of Peano Arithmetic

This work is inspired by the recent paper of Mytilinaios and Slaman [9] on the infinite injury priority method. It may be considered to fall within the general program of the study of reverse recursion theory: What axioms of Peano arithmetic are required or sufficient to prove theorems in recursion theory? Previous contributions to this program, especially with respect to the finite and infinite injury priority methods, can be found in the works of Groszek and Mytilinaios [4], Groszek and Slaman [5], Mytilinaios [8], Slaman and Woodin [10]. Results of [4] and [9], for example, together pinpoint the existence of an incomplete, nonlow r.e. degree to be provable only within some fragment of Peano arithmetic at least as strong as P- + IΣ2. Indeed an abstract principle on infinite strategies, such as that on the construction of an incomplete high r.e. degree, was introduced in [4] and shown to be equivalent to Σ2 induction over the base theory P- + IΣ0 of Peano arithmetic.

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

1990 ◽  
Vol 55 (1) ◽  
pp. 194-206 ◽  
Author(s):  
Robert S. Lubarsky
Keyword(s):  
Set Theory ◽  
Mathematical Theory ◽  
Recursion Theory ◽  
Reverse Mathematics ◽  
Peano Arithmetic ◽  
Turing Degrees ◽  
The Subject ◽  
Admissible Ordinals ◽  
Limited Induction ◽  
Well Foundedness

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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