A Uniform Characterization of Σ1-Reflection Over the Fragments of Peano Arithmetic

2021 ◽  
pp. 189-253
Author(s):  
Anton Freund
Keyword(s):  
Peano Arithmetic ◽  
Fragments Of Peano Arithmetic

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.

Lucas Against Mechanism II: A Rejoinder

1984 ◽  
Vol 14 (2) ◽  
pp. 189-191 ◽  
Author(s):  
J.R. Lucas
Keyword(s):  
David Lewis ◽  
Peano Arithmetic ◽  
Numbering Scheme

David Lewis criticizes an argument I put forward against mechansim on the grounds that I fail to distinguish between OL, Lucas's ordinary potential arithmetic output, and OML, Lucas's arithmetical output when accused of being some particular machine M; and correspondingly, between OM the ordinary potential arithmetic output of the machine M, and ONM, the arithmetic output of the machine M when accused of being a particular machine N. For any given machine, M, N, O, P, Q, R,... etc., I can in principle (my critics are often very charitable in speaking as though I could in practice, but let me revert to an ideal mind) calculate a Godel sentence for that machine - indeed infinitely many, depending on the Godel numbering scheme adopted. The Godel sentence of a particular machine can, I claim, be seen to be true, if that machine is adequate for Elementary Peano Arithmetic. Hence, if I were accused of being M, I can on that supposition see that the Godel sentence of M is true, since I am capable of Elementary Peano Arithmetic and the machine M is said to be an adequate characterization of me.

Two further combinatorial theorems equivalent to the 1-consistency of peano arithmetic

1983 ◽  
Vol 48 (4) ◽  
pp. 1090-1104 ◽  
Author(s):  
Peter Clote ◽  
Kenneth Mcaloon
Keyword(s):  
Peano Arithmetic ◽  
Infinite Subset ◽  
Linear Ordering ◽  
Tree Property ◽  
Preceding Theorem ◽  
Linear Orderings ◽  
Finite Set ◽  
The One ◽  
König’S Lemma

We give two new finite combinatorial statements which are independent of Peano arithmetic, using the methods of Kirby and Paris [6] and Paris [12]. Both are in fact equivalent over Peano arithmetic (denoted by P) to its 1-consistency. The first involves trees and the second linear orderings. Both were “motivated” by anti-basis theorems of Clote (cf. [1], [2]). The one involving trees, however, is not unrelated to the Kirby-Paris characterization of strong cuts in terms of the tree property [6], but, in fact, comes directly from König's lemma, of which it is a miniaturization. (See the remark preceding Theorem 3 below.) The resulting combinatorial statement is easily seen to imply the independent statement discovered by Mills [11], but it is not clear how to show their equivalence over Peano arithmetic without going through 1-consistency. The one involving linear orderings miniaturizes the property of infinite sets X that any linear ordering of X is isomorphic to ω or ω* on some infinite subset of X. Both statements are analogous to Example 2 of [12] and involve the notion of dense [12] or relatively large [14] finite set.We adopt the notations and definitions of [6] and [12]. We shall in particular have need of the notions of semiregular, regular and strong initial segments and of indicators.

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