Tarski’s Theorem for Arithmetic

1992 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Peano Arithmetic ◽  
Finite Alphabet ◽  
Universal Quantifier ◽  
Natural Numbers ◽  
Successor Function ◽  
Material Implication ◽  
Second Order Arithmetic ◽  
First Order ◽  
Order Arithmetic ◽  
Concrete Language

In the last chapter, we dealt with mathematical languages in considerable generality. We shall now turn to some particular mathematical languages. One of our goals is to reach Gödel’s incompleteness theorem for the particular system known as Peano Arithmetic. We shall give several proofs of this important result; the simplest one is based partly on Tarski’s theorem, to which we first turn. The first concrete language that we will study is the language of first order arithmetic based on addition, multiplication and exponentiation. [We also take as primitive the successor function and the less than or equal to relation, but these are inessential.] We shall formulate the language using only a finite alphabet (mainly for purposes of a convenient Gödel numbering); specifically we use the following 13 symbols. . . . 0’ ( ) f, υ ∽ ⊃ ∀ = ≤ # . . . The expressions 0, 0′, 0″, 0‴, · · · are called numerals and will serve as formal names of the respective natural numbers 0, 1, 2, 3, · · ·. The accent symbol (also called the prime) is serving as a name of the successor function. We also need names for the operations of addition, multiplication and exponentiation; we use the expressions f′, f″, f‴ as respective names of these three functions. We abbreviate f′ by the familiar “+”; we abbreviate f’’ by the familiar dot and f‴ by the symbol “E”. The symbols ~ and ⊃ are the familiar symbols from prepositional logic, standing for negation and material implication, respectively. [For any reader not familiar with the use of the horseshoe symbol, for any propositions p and q, the propositions p ⊃ q is intended to mean nothing more nor less than that either p is false, or p and q are both true.] The symbol ∀ is the universal quantifier and means “for all.” We will be quantifying only over natural numbers not over sets or relations on the natural numbers. [Technically, we are working in first-order arithmetic, not second-order arithmetic.] The symbol “=” is used, as usual, to denote the identity relation, and “≤” is used, as usual, to denote the “less than or equal to” relation.

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.

The mean value theorem in second order arithmetic

2001 ◽  
Vol 66 (3) ◽  
pp. 1353-1358 ◽  
Author(s):  
Christopher S. Hardin ◽  
Daniel J. Velleman
Keyword(s):  
Mean Value ◽  
Second Order ◽  
Mean Value Theorem ◽  
Natural Numbers ◽  
Second Order Arithmetic ◽  
Elementary Analysis ◽  
Logical Connectives ◽  
The Mean ◽  
Order Arithmetic ◽  
Image Position

This paper is a contribution to the project of determining which set existence axioms are needed to prove various theorems of analysis. For more on this project and its history we refer the reader to [1] and [2].We work in a weak subsystem of second order arithmetic. The language of second order arithmetic includes the symbols 0, 1, =, <, +, ·, and ∈, together with number variables x, y, z, … (which are intended to stand for natural numbers), set variables X, Y, Z, … (which are intended to stand for sets of natural numbers), and the usual quantifiers (which can be applied to both kinds of variables) and logical connectives. We write ∀x < t φ and ∃x < t φ as abbreviations for ∀x(x < t → φ) and ∃x{x < t ∧ φ) respectively; these are called bounded quantifiers. A formula is said to be if it has no quantifiers applied to set variables, and all quantifiers applied to number variables are bounded. It is if it has the form ∃xθ and it is if it has the form ∀xθ, where in both cases θ is .The theory RCA0 has as axioms the usual Peano axioms, with the induction scheme restricted to formulas, and in addition the comprehension scheme, which consists of all formulas of the formwhere φ is , ψ is , and X does not occur free in φ(n). (“RCA” stands for “Recursive Comprehension Axiom.” The reason for the name is that the comprehension scheme is only strong enough to prove the existence of recursive sets.) It is known that this theory is strong enough to allow the development of many of the basic properties of the real numbers, but that certain theorems of elementary analysis are not provable in this theory. Most relevant for our purposes is the fact that it is impossible to prove in RCA0 that every continuous function on the closed interval [0, 1] attains maximum and minimum values (see [1]).Since the most common proof of the Mean Value Theorem makes use of this theorem, it might be thought that the Mean Value Theorem would also not be provable in RCA0. However, we show in this paper that the Mean Value Theorem can be proven in RCA0. All theorems stated in this paper are theorems of RCA0, and all of our reasoning will take place in RCA0.

Proof theory

2018 ◽  
Author(s):  
Wilfried Sieg
Keyword(s):  
Finite Type ◽  
Set Theory ◽  
Proof Theory ◽  
Second Order ◽  
Von Neumann ◽  
Second Order Arithmetic ◽  
First Order ◽  
David Hilbert ◽  
Heyting Arithmetic ◽  
Order Arithmetic

Proof theory is a branch of mathematical logic founded by David Hilbert around 1920 to pursue Hilbert’s programme. The problems addressed by the programme had already been formulated, in some sense, at the turn of the century, for example, in Hilbert’s famous address to the First International Congress of Mathematicians in Paris. They were closely connected to the set-theoretic foundations for analysis investigated by Cantor and Dedekind – in particular, to difficulties with the unrestricted notion of system or set; they were also related to the philosophical conflict with Kronecker on the very nature of mathematics. At that time, the central issue for Hilbert was the ‘consistency of sets’ in Cantor’s sense. Hilbert suggested that the existence of consistent sets, for example, the set of real numbers, could be secured by proving the consistency of a suitable, characterizing axiom system, but indicated only vaguely how to give such proofs model-theoretically. Four years later, Hilbert departed radically from these indications and proposed a novel way of attacking the consistency problem for theories. This approach required, first of all, a strict formalization of mathematics together with logic; then, the syntactic configurations of the joint formalism would be considered as mathematical objects; finally, mathematical arguments would be used to show that contradictory formulas cannot be derived by the logical rules. This two-pronged approach of developing substantial parts of mathematics in formal theories (set theory, second-order arithmetic, finite type theory and still others) and of proving their consistency (or the consistency of significant sub-theories) was sharpened in lectures beginning in 1917 and then pursued systematically in the 1920s by Hilbert and a group of collaborators including Paul Bernays, Wilhelm Ackermann and John von Neumann. In particular, the formalizability of analysis in a second-order theory was verified by Hilbert in those very early lectures. So it was possible to focus on the second prong, namely to establish the consistency of ‘arithmetic’ (second-order number theory and set theory) by elementary mathematical, ‘finitist’ means. This part of the task proved to be much more recalcitrant than expected, and only limited results were obtained. That the limitation was inevitable was explained in 1931 by Gödel’s theorems; indeed, they refuted the attempt to establish consistency on a finitist basis – as soon as it was realized that finitist considerations could be carried out in a small fragment of first-order arithmetic. This led to the formulation of a general reductive programme. Gentzen and Gödel made the first contributions to this programme by establishing the consistency of classical first-order arithmetic – Peano arithmetic (PA) – relative to intuitionistic arithmetic – Heyting arithmetic. In 1936 Gentzen proved the consistency of PA relative to a quantifier-free theory of arithmetic that included transfinite recursion up to the first epsilon number, ε0; in his 1941 Yale lectures, Gödel proved the consistency of the same theory relative to a theory of computable functionals of finite type. These two fundamental theorems turned out to be most important for subsequent proof-theoretic work. Currently it is known how to analyse, in Gentzen’s style, strong subsystems of second-order arithmetic and set theory. The first prong of proof-theoretic investigations, the actual formal development of parts of mathematics, has also been pursued – with a surprising result: the bulk of classical analysis can be developed in theories that are conservative over (fragments of) first-order arithmetic.

On generalized quantifiers in arithmetic

1982 ◽  
Vol 47 (1) ◽  
pp. 187-190 ◽  
Author(s):  
Carl Morgenstern
Keyword(s):  
Axiom System ◽  
Peano Arithmetic ◽  
Order Logic ◽  
First Order Logic ◽  
Generalized Quantifiers ◽  
First Order ◽  
Order Language ◽  
Truth Definition ◽  
Order Arithmetic ◽  
Infinite Set

In this note we investigate an extension of Peano arithmetic which arises from adjoining generalized quantifiers to first-order logic. Markwald [2] first studied the definability properties of L1, the language of first-order arithmetic, L, with the additional quantifer Ux which denotes “there are infinitely many x such that…. Note that Ux is the same thing as the Keisler quantifier Qx in the ℵ0 interpretation.We consider L2, which is L together with the ℵ0 interpretation of the Magidor-Malitz quantifier Q2xy which denotes “there is an infinite set X such that for distinct x, y ∈ X …”. In [1] Magidor and Malitz presented an axiom system for languages which arise from adding Q2 to a first-order language. They proved that the axioms are valid in every regular interpretation, and, assuming ◊ω1, that the axioms are complete in the ℵ1 interpretation.If we let denote Peano arithmetic in L2 with induction for L2 formulas and the Magidor-Malitz axioms as logical axioms, we show that in we can give a truth definition for first-order Peano arithmetic, . Consequently we can prove in that is Πn sound for every n, thus in we can prove the Paris-Harrington combinatorial principle and the higher-order analogues due to Schlipf.

Decidability and undecidability of theories with a predicate for the primes

1993 ◽  
Vol 58 (2) ◽  
pp. 672-687 ◽  
Author(s):  
P. T. Bateman ◽  
C. G. Jockusch ◽  
A. R. Woods
Keyword(s):  
General Result ◽  
Order Theory ◽  
Second Order ◽  
The Other ◽  
Linear Case ◽  
Natural Numbers ◽  
Successor Function ◽  
First Order ◽  
First Order Theory

AbstractIt is shown, assuming the linear case of Schinzel's Hypothesis, that the first-order theory of the structure 〈ω; +, P〉, where P is the set of primes, is undecidable and, in fact, that multiplication of natural numbers is first-order definable in this structure. In the other direction, it is shown, from the same hypothesis, that the monadic second-order theory of 〈ω S, P〉 is decidable, where S is the successor function. The latter result is proved using a general result of A. L. Semënov on decidability of monadic theories, and a proof of Semënov's result is presented.

IN GOOD COMPANY? ON HUME’S PRINCIPLE AND THE ASSIGNMENT OF NUMBERS TO INFINITE CONCEPTS

2015 ◽  
Vol 8 (2) ◽  
pp. 370-410 ◽  
Author(s):  
PAOLO MANCOSU
Keyword(s):  
Nineteenth Century ◽  
Recent Article ◽  
Natural Numbers ◽  
Hume’S Principle ◽  
Second Order Arithmetic ◽  
The Third ◽  
Infinite Sets ◽  
Order Arithmetic ◽  
Countably Infinite ◽  
Abstraction Principles

AbstractIn a recent article (Mancosu, 2009), I have explored the historical, mathematical, and philosophical issues related to the new theory of numerosities. The theory of numerosities provides a context in which to assign ‘sizes’ to infinite sets of natural numbers in such a way as to preserve the part-whole principle, namely if a setAis properly included inBthen the numerosity ofAis strictly less than the numerosity ofB. Numerosity assignments differ from the standard assignment of size provided by Cantor’s cardinality assignments. In this paper I generalize some worries, raised by Richard Heck, emerging from the theory of numerosities to a line of thought resulting in what I call a ‘good company’ objection to Hume’s Principle (HP). The paper is centered around five main parts. The first (§3) takes a historical look at nineteenth-century attributions of equality of numbers in terms of one-one correlation and argues that there was no agreement as to how to extend such determinations to infinite sets of objects. This leads to the second part (§4) where I show that there are countably-infinite many abstraction principles that are ‘good’, in the sense that they share the same virtues of HP (or so I claim) and from which we can derive the axioms of second-order arithmetic. All the principles I present agree with HP in the assignment of numbers to finite concepts but diverge from it in the assignment of numbers to infinite concepts. The third part (§5) connects this material to a debate on Finite Hume’s Principle between Heck and MacBride. The fourth part (§6) states the ‘good company’ objection as a generalization of Heck’s objection to the analyticity of HP based on the theory of numerosities. In the same section I offer a taxonomy of possible neo-logicist responses to the ‘good company’ objection. Finally, §7 makes a foray into the relevance of this material for the issue of cross-sortal identifications for abstractions.

Forcing and reducibilities. III. Forcing in fragments of set theory

1983 ◽  
Vol 48 (4) ◽  
pp. 1013-1034
Author(s):  
Piergiorgio Odifreddi
Keyword(s):  
Set Theory ◽  
Recursion Theory ◽  
Second Order ◽  
Analytic Hierarchy ◽  
Second Order Arithmetic ◽  
First Order ◽  
Order Language ◽  
Order Arithmetic ◽  
Definition Of ◽  
Generic Set

We conclude here the treatment of forcing in recursion theory begun in Part I and continued in Part II of [31]. The numbering of sections is the continuation of the numbering of the first two parts. The bibliography is independent.In Part I our language was a first-order language: the only set we considered was the (set constant for the) generic set. In Part II a second-order language was introduced, and we had to interpret the second-order variables in some way. What we did was to consider the ramified analytic hierarchy, defined by induction as:A0 = {X ⊆ ω: X is arithmetic},Aα+1 = {X ⊆ ω: X is definable (in 2nd order arithmetic) over Aα},Aλ = ⋃α<λAα (λ limit),RA = ⋃αAα.We then used (a relativized version of) the fact that (Kleene [27]). The definition of RA is obviously modeled on the definition of the constructible hierarchy introduced by Gödel [14]. For this we no longer work in a language for second-order arithmetic, but in a language for (first-order) set theory with membership as the only nonlogical relation:L0 = ⊘,Lα+1 = {X: X is (first-order) definable over Lα},Lλ = ⋃α<λLα (λ limit),L = ⋃αLα.

scholarly journals A simple model for a weak system of arithmetic

1974 ◽  
Vol 11 (3) ◽  
pp. 321-323
Author(s):  
D.W. Barnes ◽  
G.P. Monro
Keyword(s):  
Simple Model ◽  
Complete List ◽  
Natural Numbers ◽  
Successor Function ◽  
First Order ◽  
Nonstandard Model ◽  
Infinite Elements

The natural, first order version of Peano's axioms (the theory T with 0, the successor function and an induction schema) is shown to possess the following nonstandard model: the natural numbers together with a collection of ‘infinite’ elements isomorphic to the integers. In fact, a complete list of the models of this theory is obtained by showing that T is equivalent to the apparently weaker theory with the induction axiom replaced by axioms stating that there are no finite cycles under the successor function and that 0 is the only non-successor.

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.

An infinitistic rule of proof

1968 ◽  
Vol 32 (4) ◽  
pp. 447-451 ◽  
Author(s):  
H. B. Enderton
Keyword(s):  
Formal System ◽  
Peano Arithmetic ◽  
Second Order ◽  
Second Order Arithmetic ◽  
Order Arithmetic

In this paper we consider a fonnal system of second-order Peano arithmetic with a rule of inference stronger than the ω-rule [3]. We also consider the relation to a class of models for analysis (i.e. second-order arithmetic) which lies between the class of ω-models and the class of β-models [5].The notation used is largely that of [3] and [5]. We assume that the reader has some familiarity with at least the ideas of the former. The formal system (A) of Peano arithmetic employed in [3] includes the comprehension axioms and the second-order induction axiom.

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