Arithmetical Comprehension

2019 ◽  
pp. 109-129
Author(s):  
John Stillwell
Keyword(s):  
Peano Arithmetic ◽  
Sweet Spot ◽  
Natural Numbers ◽  
Axiom Schema ◽  
Axiom Systems ◽  
Comprehension Axiom

This chapter focuses on arithmetical comprehension. Arithmetical comprehension is the most obvious set existence axiom to use when developing analysis in a system based on Peano arithmetic (PA) with set variables. This axiom asserts the existence of a set X of natural numbers for each property φ‎ definable in the language of PA. More precisely, if φ‎(n) is a property defined in the language of PA plus set variables, but with no set quantifiers, then there is a set X whose members are the natural numbers n such that φ‎(n). Since all such formulas φ‎ are asserted for, the arithmetical comprehension axiom is really an axiom schema. The reason set variables are allowed in φ‎ is to enable sets to be defined in terms of “given” sets. The reason set quantifiers are disallowed in φ‎ is to avoid definitions in which a set is defined in terms of all sets of natural numbers (and hence in terms of itself). The system consisting of PA plus arithmetical comprehension is called ACA0. This system lies at a remarkable “sweet spot” among axiom systems for analysis.

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.

Theories without countable models

1972 ◽  
Vol 37 (3) ◽  
pp. 562-568
Author(s):  
Andreas Blass
Keyword(s):  
Standard Model ◽  
Countable Model ◽  
Compactness Theorem ◽  
Complete Theory ◽  
Natural Numbers ◽  
Axiom Schema ◽  
First Order ◽  
Order Language ◽  
The Standard Model ◽  
Skolem Theorem

Consider the Löwenheim-Skolem theorem in the form: If a theory in a countable first-order language has a model, then it has a countable model. As is well known, this theorem becomes false if one omits the hypothesis that the language be countable, for one then has the following trivial counterexample.Example 1. Let the language have uncountably many constants, and let the theory say that they are unequal.To motivate some of our future definitions and to introduce some notation, we present another, less trivial, counterexample.Example 2. Let L0 be the language whose n-place predicate (resp. function) symbols are all the n-place predicates (resp. functions) on the set ω of natural numbers. Let be the standard model for L0; we use the usual notation Th() for its complete theory. Add to L0 a new constant e, and add to Th() an axiom schema saying that e is infinite. By the compactness theorem, the resulting theory T has models. However, none of its models are countable. Although this fact is well known, we sketch a proof in order to refer to it later.By [5, p. 81], there is a family {Aα ∣ < α < c} of infinite subsets of ω, the intersection of any two of which is finite.

Independent Gödel sentences and independent sets

1975 ◽  
Vol 40 (2) ◽  
pp. 159-166
Author(s):  
A. M. Dawes ◽  
J. B. Florence
Keyword(s):  
Positive Integer ◽  
Independent Sets ◽  
Peano Arithmetic ◽  
Natural Numbers ◽  
Simple Set ◽  
Logical Notion ◽  
Infinite Set ◽  
Element Set

In this paper we investigate some of the recursion-theoretic problems which are suggested by the logical notion of independence.A set S of natural numbers will be said to be k-independent (respectively, ∞-independent) if, roughly speaking, in every correct system there is a k-element set (respectively, an infinite set) of independent true sentences of the form x ∈ S. S will be said to be effectively independent (respectively, absolutely independent) if given any correct system we can generate an infinite set of independent (respectively, absolutely independent) true sentences of the form x ∈ S.We prove that(a) S is absolutely independent ⇔S is effectively independent ⇔S is productive;(b) for every positive integer k there is a Π1 set which is k-independent but not (k + 1)-independent;(c) there is a Π1 set which is k-independent for all k but not ∞-independent;(d) there is a co-simple set which is ∞-independent.We also give two new proofs of the theorem of Myhill [1] on the existence of an infinite set of Σ1 sentences which are absolutely independent relative to Peano arithmetic. The first proof uses the existence of an absolutely independent Π1 set of natural numbers, and the second uses a modification of the method of Gödel and Rosser.

Internal categoricity and the natural numbers

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Peano Arithmetic ◽  
Higher Order ◽  
Order Logic ◽  
Natural Numbers ◽  
Important Work ◽  
Higher Order Logic

The simple conclusion of the preceding chapters is that moderate modelism fails. But this leaves us with a choice between abandoning moderation and abandoning modelism. The aim of this chapter, and the next couple of chapters, is to outline a speculative way to save moderation by abandoning modelism. The idea is to do metamathematics without semantics, by working deductively in a higher-order logic. In this chapter, the focus is on the internal categoricity of arithmetic. After formalising an internal notion of a model of the Peano axioms, we show how to internalise Dedekind’s Categority Theorem. The resulting “intolerance” of Peano arithmetic provides internalists with a way to draw the distinction between algebraic and univocal theories. In the appendices, we discuss how this relates to Parsons’ important work, and establish a certain dependence of the internal categoricity theorem on higher-order logic.

Arithmetic and the theory of types

1984 ◽  
Vol 49 (2) ◽  
pp. 621-624 ◽  
Author(s):  
M. Boffa
Keyword(s):  
Peano Arithmetic ◽  
Positive Answer ◽  
Second Order ◽  
Countable Model ◽  
Natural Numbers ◽  
Conservative Extension ◽  
Logical Definition ◽  
Claude Bernard ◽  
Definition Of ◽  
Image Position

A hundred years ago, Frege proposed a logical definition of the natural numbers based on the following idea:He replaced this circular definition by the following one:He tried afterwards to found his theory over a notion of class satisfying a general comprehension principle:Russell quickly derived a contradiction from this principle (the famous Russell's paradox) but saved Frege's arithmetic with his theory of types based on the following comprehension principle:In 1979, talking at the Claude Bernard University in Lyon, I remarked that 3 types suffice to provide Frege's arithmetic, showing in fact that PA2 (second order Peano arithmetic) holds in TT3 + AI (theory of types 0, 1, 2 plus a suitable axiom of infinity). I asked whether TT3 + AI was a conservative extension of PA2. Pabion [3] gave a positive answer by a subtle use of the Fraenkel-Moskowski method. This result will be improved in the present paper, with a view to getting models of NF3 + AI in which Frege's arithmetic forms a model isomorphic to a given countable model of PA2.

scholarly journals Multi-sorted version of second order arithmetic

2016 ◽  
Vol 13 (5) ◽  
Author(s):  
Farida Kachapova
Keyword(s):  
Reverse Mathematics ◽  
Peano Arithmetic ◽  
Second Order ◽  
Natural Numbers ◽  
Natural Form ◽  
Second Order Arithmetic ◽  
Arithmetical Truth ◽  
Order Arithmetic

This paper describes axiomatic theories SA and SAR, which are versions of second order arithmetic with countably many sorts for sets of natural numbers. The theories are intended to be applied in reverse mathematics because their multi-sorted language allows to express some mathematical statements in more natural form than in the standard second order arithmetic. We study metamathematical properties of the theories SA, SAR and their fragments. We show that SA is mutually interpretable with the theory of arithmetical truth PATr obtained from the Peano arithmetic by adding infinitely many truth predicates. Corresponding fragments of SA and PATr are also mutually interpretable. We compare the proof-theoretical strengths of the fragments; in particular, we show that each fragment SAs with sorts <=s is weaker than next fragment SAs+1.

scholarly journals Bi-Modal Naive Set Theory

2018 ◽  
Vol 15 (2) ◽  
pp. 139
Author(s):  
John Wigglesworth
Keyword(s):  
Set Theory ◽  
Kripke Model ◽  
Axiom Schema ◽  
Modal Operators ◽  
Modal Theory ◽  
Naive Set Theory ◽  
Naïve Comprehension ◽  
Comprehension Axiom ◽  
Forward Looking

This paper describes a modal conception of sets, according to which sets are 'potential' with respect to their members.  A modal theory is developed, which invokes a naive comprehension axiom schema, modified by adding `forward looking' and `backward looking' modal operators.  We show that this `bi-modal' naive set theory can prove modalized interpretations of several ZFC axioms, including the axiom of infinity.  We also show that the theory is consistent by providing an S5 Kripke model.  The paper concludes with some discussion of the nature of the modalities involved, drawing comparisons with noneism, the view that there are some non-existent objects.

Remarks on Peano Arithmetic

2000 ◽  
Vol 20 (1) ◽  
Author(s):  
Charles Sayward
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Peano Arithmetic ◽  
The Other ◽  
Natural Numbers ◽  
Recursive Definition ◽  
Explicit Definition

<p>Russell held that the theory of natural numbers could be derived from three primitive concepts: number, successor and zero. This leaves out multiplication and addition. Russell introduces these concepts by recursive definition. It is argued that this does not render addition or multiplication any less primitive than the other three. To this it might be replied that any recursive definition can be transformed into a complete or explicit definition with the help of a little set theory. But that is a point about set theory, not number theory. We have learned more about the distinction between logic and set theory than was known in Russell's day, especially as this affects logicist aspirations.</p>

scholarly journals PROOF-THEORETIC STRENGTHS OF WEAK THEORIES FOR POSITIVE INDUCTIVE DEFINITIONS

2018 ◽  
Vol 83 (3) ◽  
pp. 1091-1111 ◽  
Author(s):  
TOSHIYASU ARAI
Keyword(s):  
Transfinite Induction ◽  
Boundary Line ◽  
Natural Numbers ◽  
Inductive Definitions ◽  
Image Position ◽  
Weak Theories ◽  
Comprehension Axiom

AbstractIn this article the lightface ${\rm{\Pi }}_1^1$-Comprehension axiom is shown to be proof-theoretically strong even over ${\rm{RCA}}_0^{\rm{*}}$, and we calibrate the proof-theoretic ordinals of weak fragments of the theory ${\rm{I}}{{\rm{D}}_1}$ of positive inductive definitions over natural numbers. Conjunctions of negative and positive formulas in the transfinite induction axiom of ${\rm{I}}{{\rm{D}}_1}$ are shown to be weak, and disjunctions are strong. Thus we draw a boundary line between predicatively reducible and impredicative fragments of ${\rm{I}}{{\rm{D}}_1}$.

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.

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