Some General Remarks on Provability and Truth

1992 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Peano Arithmetic ◽  
Mathematical Induction ◽  
Natural Numbers ◽  
Original Proof ◽  
The Third ◽  
Consistent Extension ◽  
Order Arithmetic ◽  
Full Force ◽  
Expressible Set ◽  
Principle Of Mathematical Induction

We have given three different incompleteness proofs of Peano Arithmetic— the first used Tarski’s truth-set, the second (Gödel’s original proof) was based on the assumption of ω-consistency, and the third (Rosser’s proof) was based on the assumption of simple consistency. The three proofs yield different generalizations—namely 1. Every axiomatizable subsystem of N is incomplete. 2. Every axiomatizable ω-consistent system in which all true Σ0-sentences are provable is incomplete. 3. Every axiomatizable simply consistent extension of (R) is incomplete. The first of the three proofs is by far the simplest and we are surprised that it has not appeared in more textbooks. Of course, it can be criticized on the grounds that it is not formalizable in arithmetic (since the truth set is not expressible in arithmetic), but this should be taken with some reservations in light of Askanas’ theorem, which we will discuss a bit later. It is not too surprising that Peano Arithmetic is incomplete because the scheme of mathematical induction does not really express the full force of mathematical induction. The true principle of mathematical induction is that for any set A of natural numbers, if A contains 0 and A is closed under the successor function (such a set A is sometimes called an inductive set), then A contains all natural numbers. Now, there are non-denumerably many sets of natural numbers but only denumerably many formulas in the language LA and, hence, there are only denumerably many expressible sets of LA- Therefore, the formal axiom scheme of induction for P.A. guarantees only that for every expressible set A, if A is inductive, then A contains all natural numbers. To express the principle of mathematical induction fully, we need second order arithmetic in which we take set and relational variables and quantify over sets and relations of natural numbers.

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.

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.

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.

A system of completely independent axioms for the sequence of natural numbers

1943 ◽  
Vol 8 (2) ◽  
pp. 41-44 ◽  
Author(s):  
Shianghaw Wang
Keyword(s):  
Mathematical Induction ◽  
Natural Numbers ◽  
Complete Independence ◽  
Principle Of Mathematical Induction ◽  
Elegant Form

Peano's five axioms for the sequence of natural numbers run as follows: (1) 1 is a number, (2) Any number α has a unique sequent a′ (3) Different numbers have different sequents, (4) 1 is not a sequent, and (5) The principle of mathematical induction.Now, in a system of axioms, it seems desirable, in the first place, to avoid symbols for particular individuals and symbols for functions, and, secondly, to secure the property of complete independence. In fact, Peano's system can be replaced by an equivalent one which fulfills both requirements. In such a system, the symbol “1” and the function “sequent” will not appear, although they may be introduced afresh through definitions. Besides, the principle of mathematical induction will no more play the rôle of an axiom, because, under the second requirement, it can hardly be stated in a simple and elegant form.

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.

On the role of Ramsey quantifiers in first order arithmetic

1982 ◽  
Vol 47 (2) ◽  
pp. 423-435 ◽  
Author(s):  
James H. Schmerl ◽  
Stephen G. Simpson
Keyword(s):  
Formal System ◽  
Intended Meaning ◽  
Completeness Proof ◽  
First Order ◽  
Uncountable Cardinal ◽  
Consistent Extension ◽  
Witness Set ◽  
Order Arithmetic ◽  
Infinite Set

The purpose of this paper is to study a formal system PA(Q2) of first order Peano arithmetic, PA, augmented by a Ramsey quantifier Q2 which binds two free variables. The intended meaning of Q2xx′φ(x, x′) is that there exists an infinite set X of natural numbers such that φ(a, a′) holds for all a, a′ Є X such that a ≠ a′. Such an X is called a witness set for Q2xx′φ(x, x′). Our results would not be affected by the addition of further Ramsey quantifiers Q3, Q4, …, Here of course the intended meaning of Qkx1 … xkφ(x1,…xk) is that there exists an infinite set X such that φ(a1…, ak) holds for all k-element subsets {a1, … ak} of X.Ramsey quantifiers were first introduced in a general model theoretic setting by Magidor and Malitz [13]. The system PA{Q2), or rather, a system essentially equivalent to it, was first defined and studied by Macintyre [12]. Some of Macintyre's results were obtained independently by Morgenstern [15]. The present paper is essentially self-contained, but all of our results have been directly inspired by those of Macintyre [12].After some preliminaries in §1, we begin in §2 by giving a new completeness proof for PA(Q2). A by-product of our proof is that for every regular uncountable cardinal k, every consistent extension of PA(Q2) has a k-like model in which all classes are definable. (By a class we mean a subset of the universe of the model, every initial segment of which is finite in the sense of the model.)

Arithmetic, philosophical issues in

2018 ◽  
Author(s):  
Michael Potter
Keyword(s):  
Natural Number ◽  
Decision Procedure ◽  
Mathematical Induction ◽  
Universal Quantifier ◽  
Number Concept ◽  
Special Character ◽  
The Status ◽  
Proof By Induction ◽  
Philosophy Of Arithmetic ◽  
Principle Of Mathematical Induction

The philosophy of arithmetic gains its special character from issues arising out of the status of the principle of mathematical induction. Indeed, it is just at the point where proof by induction enters that arithmetic stops being trivial. The propositions of elementary arithmetic – quantifier-free sentences such as ‘7+5=12’ – can be decided mechanically: once we know the rules for calculating, it is hard to see what mathematical interest can remain. As soon as we allow sentences with one universal quantifier, however – sentences of the form ‘(∀x)f(x)=0’ – we have no decision procedure either in principle or in practice, and can state some of the most profound and difficult problems in mathematics. (Goldbach’s conjecture that every even number greater than 2 is the sum of two primes, formulated in 1742 and still unsolved, is of this type.) It seems natural to regard as part of what we mean by natural numbers that they should obey the principle of induction. But this exhibits a form of circularity known as ‘impredicativity’: the statement of the principle involves quantification over properties of numbers, but to understand this quantification we must assume a prior grasp of the number concept, which it was our intention to define. It is nowadays a commonplace to draw a distinction between impredicative definitions, which are illegitimate, and impredicative specifications, which are not. The conclusion we should draw in this case is that the principle of induction on its own does not provide a non-circular route to an understanding of the natural number concept. We therefore need an independent argument. Four broad strategies have been attempted, which we shall consider in turn.

Joachimist Influences on the Idea of a Last World Emperor

Traditio ◽  
1961 ◽  
Vol 17 ◽  
pp. 323-370 ◽  
Author(s):  
Marjorie Reeves
Keyword(s):  
Sixteenth Century ◽  
Middle Ages ◽  
Holy Spirit ◽  
Time Process ◽  
Third Age ◽  
The Third ◽  
Last Judgment ◽  
The Holy Spirit ◽  
Christian Thought ◽  
Full Force

The question of the dramatis personae in the last great act of history was a subject of perennial interest in the Middle Ages. Parts, both good and bad, had to be cast and it is not surprising that national hopes and rivalries frequently crept into the various attempts to assign these tremendous cosmic roles. Although both the pessimistic expectation of a mounting crescendo of evil and the hope of a millennium had existed in Christian thought since its beginning, it was the Joachimist structure of history which most clearly brought together the final crisis of evil and the final blessedness in a last great act which was yet within history, separated from eternity by the Second Advent. The concept of an age of blessedness had three strands in it: first, the idea of the millennium, derived from the Apocalypse (20.1–3), in which Satan is bound for a thousand years; secondly, the concept of a Sabbath Age, symbolized in the Seventh Day of Creation when God rested from His labors; thirdly, the Trinitarian interpretation of history, finally worked out by Joachim, in which history was expected to culminate in the Third Age of the Holy Spirit. The first two ideas did not necessarily lead to the expectation of a last age of blessedness within time: the millennium was frequently interpreted as covering the whole period between the First and Second Advents, or again, as constituting a rule of Christ and His Saints beyond history; the Sabbath Age could be seen as a Sabbath beyond the Second Advent and Last Judgment and therefore also beyond history. It was only when these two concepts became linked with the Trinitarian view of history that they clearly symbolized a crowning age of history, set in the future and therefore not yet attained, whilst unmistakably within the time process, preceding the winding-up of history in the Second Advent and Last Judgment. The full force of Joachim's concept of the Third Age was rarely grasped, appearing usually in a much-debased form, but the program of Last Things, as worked out by Joachites of the thirteenth and fourteenth centuries, profoundly influenced the form which these expectations took in the later Middle Ages and, indeed, right down to the end of the sixteenth century.

PROVABILITY LOGICS RELATIVE TO A FIXED EXTENSION OF PEANO ARITHMETIC

2018 ◽  
Vol 83 (3) ◽  
pp. 1229-1246
Author(s):  
TAISHI KURAHASHI
Keyword(s):  
Peano Arithmetic ◽  
Provability Logic ◽  
Consistent Extension ◽  
Image Position ◽  
Computably Enumerable

AbstractLet T and U be any consistent theories of arithmetic. If T is computably enumerable, then the provability predicate $P{r_\tau }\left( x \right)$ of T is naturally obtained from each ${{\rm{\Sigma }}_1}$ definition $\tau \left( v \right)$ of T. The provability logic $P{L_\tau }\left( U \right)$ of τ relative to U is the set of all modal formulas which are provable in U under all arithmetical interpretations where □ is interpreted by $P{r_\tau }\left( x \right)$. It was proved by Beklemishev based on the previous studies by Artemov, Visser, and Japaridze that every $P{L_\tau }\left( U \right)$ coincides with one of the logics $G{L_\alpha }$, ${D_\beta }$, ${S_\beta }$, and $GL_\beta ^ -$, where α and β are subsets of ω and β is cofinite.We prove that if U is a computably enumerable consistent extension of Peano Arithmetic and L is one of $G{L_\alpha }$, ${D_\beta }$, ${S_\beta }$, and $GL_\beta ^ -$, where α is computably enumerable and β is cofinite, then there exists a ${{\rm{\Sigma }}_1}$ definition $\tau \left( v \right)$ of some extension of $I{{\rm{\Sigma }}_1}$ such that $P{L_\tau }\left( U \right)$ is exactly L.

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