Weak partition relations, finite games, and independence results in Peano arithmetic

1980 ◽  
pp. 92-107 ◽  
Author(s):  
Peter Clote
Keyword(s):  
Peano Arithmetic ◽  
Finite Games ◽  
Partition Relations ◽  
Independence Results

Some independence results for Peano arithmetic

1978 ◽  
Vol 43 (4) ◽  
pp. 725-731 ◽  
Author(s):  
J. B. Paris
Keyword(s):  
Significant Contribution ◽  
Peano Arithmetic ◽  
Natural Numbers ◽  
Theoretic Method ◽  
First Order ◽  
Detailed Proof ◽  
Finite Games ◽  
Independence Results ◽  
First Time

In this paper we shall outline a purely model theoretic method for obtaining independence results for Peano's first order axioms (P). The method is of interest in that it provides for the first time elementary combinatorial statements about the natural numbers which are not provable in P. We give several examples of such statements.Central to this exposition will be the notion of an indicator. Indicators were introduced by L. Kirby and the author in [3] although they had occurred implicitly in earlier papers, for example Friedman [1]. The main result on indicators which we shall need (Lemma 1) was proved by Laurie Kirby and the author in the summer of 1976 but it was not until early in the following year that the author realised that this lemma could be used to give independence results.The first combinatorial independence results obtained were essentially statements about certain finite games and consequently were not immediately meaningful (see Example 2). This shortcoming was remedied by Leo Harrington who, upon hearing an incorrect version of our results, noticed a beautifully simply independent combinatorial statement. We outline this result in Example 3. An alternative, more detailed, proof may be found in [5].Clearly Laurie Kirby and Leo Harrington have made a very significant contribution to this paper and we wish to express our sincere thanks to them.

scholarly journals On the induction schema for decidable predicates

2003 ◽  
Vol 68 (1) ◽  
pp. 17-34 ◽  
Author(s):  
Lev D. Beklemishev
Keyword(s):  
Open Problem ◽  
Inference Rule ◽  
Peano Arithmetic ◽  
Induction Principle ◽  
Independence Results ◽  
Computable Functions

AbstractWe study the fragment of Peano arithmetic formalizing the induction principle for the class of decidable predicates, IΔ1. We show that IΔ1 is independent from the set of all true arithmetical Π2-sentences. Moreover, we establish the connections between this theory and some classes of oracle computable functions with restrictions on the allowed number of queries. We also obtain some conservation and independence results for parameter free and inference rule forms of Δ1-induction.An open problem formulated by J. Paris (see [4, 5]) is whether IΔ1 proves the corresponding least element principle for decidable predicates, LΔ1 (or, equivalently, the Σ1-collection principle BΣ1). We reduce this question to a purely computation-theoretic one.

NEGATIVE PARTITION RELATIONS OR UNCOUNTABLE CARDINALS OF COFINALITY

2001 ◽  
Vol 37 (1-2) ◽  
pp. 233-236
Author(s):  
P. Matet
Keyword(s):  
Partition Relations ◽  
Uncountable Cardinals

We modify an argument of Baumgartner to show that…

The Natural Numbers

2018 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Natural Number ◽  
Peano Arithmetic ◽  
Number Sequence ◽  
Natural Numbers ◽  
Ordinal Numbers ◽  
Cardinal Numbers

How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, the natural numbers are cardinal numbers, individuated by the cardinalities of the collections that they number. Another answer regards the natural numbers as ordinal numbers, individuated by their positions in the natural number sequence. Some reasons to favor the second answer are presented. This answer is therefore developed in more detail, involving a form of abstraction on numerals. Based on this answer, a justification for the axioms of Dedekind–Peano arithmetic is developed.

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