Theory Of Natural Numbers Research Articles

Infinity and continuum in the alternative set theory

Alternative set theory was created by the Czech mathematician Petr Vopěnka in 1979 as an alternative to Cantor’s set theory. Vopěnka criticised Cantor’s approach for its loss of correspondence with the real world. Alternative set theory can be partially axiomatised and regarded as a nonstandard theory of natural numbers. However, its intention is much wider. It attempts to retain a correspondence between mathematical notions and phenomena of the natural world. Through infinity, Vopěnka grasps the phenomena of vagueness. Infinite sets are defined as sets containing proper semisets, i.e. vague parts of sets limited by the horizon. The new interpretation extends the field of applicability of mathematics and simultaneously indicates its limits. Compared to strict finitism and other attempts at a reduction of the infinite to the finite Vopěnka’s theory reverses the process: he models the finite in the infinite.

Multi-dimensional Interpretations of Presburger Arithmetic in Itself

Abstract Presburger arithmetic is the true theory of natural numbers with addition. We study interpretations of Presburger arithmetic in itself. The main result of this paper is that all self-interpretations are definably isomorphic to the trivial one. Here we consider interpretations that might be multi-dimensional. We note that this resolves a conjecture by Visser (1998, An overview of interpretability logic. Advances in Modal Logic, pp. 307–359). In order to prove the result, we show that all linear orderings that are interpretable in $({\mathbb{N}},+)$ are scattered orderings with the finite Hausdorff rank and that the ranks are bounded in the terms of the dimensions of the respective interpretations.

Некоторые проблемы разрешимости и выразимости для предиката делимости на два последовательных числа

The predicate of divisibility on two consecutive numbers DW(x,y) ÷ x | y ∧ 1+x | y was introduced by L. van den Dries and A. Wilkie when they studied some properties of subsets of natural numbers, existentially definable with unit, addition and divisibility. Undecidability of the existential theory of natural numbers with multiplication and DW and definability of addition and multiplication in terms of DWand divisibility is proved in the paper. Then the definability of multiplication in the terms of addition and DW is proved. Some definability questions for order and DW are also considered in the paper.

重建自然数理论

重建自然数理论

A Simple Character String Proof of the "True but Unprovable'' Version of Gödel's First Incompleteness Theorem

A rather easy yet rigorous proof of a version of G\"odel's first incompleteness theorem is presented. The version is "each recursively enumerable theory of natural numbers with 0, 1, +, *, =, logical and, logical not, and the universal quantifier either proves a false sentence or fails to prove a true sentence". The proof proceeds by first showing a similar result on theories of finite character strings, and then transporting it to natural numbers, by using them to model strings and their concatenation. Proof systems are expressed via Turing machines that halt if and only if their input string is a theorem. This approach makes it possible to present all but one parts of the proof rather briefly with simple and straightforward constructions. The details require some care, but do not require significant background knowledge. The missing part is the widely known fact that Turing machines can perform complicated computational tasks.

Scheme-based theorem discovery and concept invention

We describe an approach to automatically invent/explore new mathematical theories, with the goal of producing results comparable to those produced by humans, as represented, for example, in the libraries of the Isabelle proof assistant. Our approach is based on ‘schemes’, which are formulae in higher-order logic. We show that it is possible to automate the instantiation process of schemes to generate conjectures and definitions. We also show how the new definitions and the lemmata discovered during the exploration of a theory can be used, not only to help with the proof obligations during the exploration, but also to reduce redundancies inherent in most theory-formation systems. We exploit associative-commutative (AC) operators using ordered rewriting to avoid AC variations of the same instantiation. We implemented our ideas in an automated tool, called IsaScheme, which employs Knuth–Bendix completion and recent automatic inductive proof tools. We have evaluated our system in a theory of natural numbers and a theory of lists.

O "salto arquimediano": um processo de ruptura epistemológica no pensamento matemático

In this paper we introduce the concept of as a non deductive argumentation process in mathematics which produces an epistemological rupture in the direction of giving objectivity to certain mathematical facts. We also discuss the qualitative character of archimedian leap emphasizing its hermeneutical function in mathematical thought and its relation with an aesthetical knowledge in mathematics. Some examples of epistemological ruptures produced by an archimedian leap, among others, are the following: (1) the archimedian property of the real line, which structures the Euclidean line through the real number system; (2) the complete induction principle of the theory of natural numbers, which models our intuition about recursive processes; (3) the passage of potential infinite to actual infinite, which permits giving an objective content to the theory of infinite sets and cardinalities.

The development of students’ understanding of the numerical value of fractions

An experiment is reported that investigated the development of students’ understanding of the numerical value of fractions. A total of 200 students ranging in age from 10 to 16 years were tested using a questionnaire that required them to decide on the smallest/biggest fraction, to order a set of given fractions and to justify their responses. Students’ responses were grouped in categories that revealed three main explanatory frameworks within which fractions seem to be interpreted. The first explanatory framework, emerging directly from the initial theory of natural numbers, is that fraction consists of two independent numbers. The second considers fractions as parts of a whole. Only in the third explanatory framework, students were able to understand the relation between numerator and denominator and to consider that fractions can be smaller, equal or even bigger than the unit.

Collapse Result for Extensions of the Presburger Arithmetic by a Unary Function Compatible with Addition

Earlier, Belegradek, Stolboushkin, and Taitslin proved that the collapse result holds in the theory of natural numbers with addition, i.e., each locally generic query using addition can be written without it. In this paper, we use the sufficient conditions of the collapse result obtained by Taitslin to prove that it holds in any extensions of the Presburger arithmetic by a unary function compatible with addition. The notion of a function compatible with addition was proposed by A. L. Semenov.

Construction of an Automaton From a Formula of the Monadic First-Order Theory of Natural Numbers

A method for construction of an automaton (an acceptor) from a first-order logical formula is proposed. The method is similar to the tableau method and can be considered as modification and improvement of it.

Remarks on Peano Arithmetic

<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>

Remarks on Peano Arithmetic

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.

Simple sentences that are hard to decide

It was considered to be “typical for first order theories” that a restriction to sentences with only a limited number of quantifier alternations leads to an exponential decrease of complexity. Using domino games, which were treated in a previous paper to describe computations of alternating Turing machines, we prove that this is not always true. We present a list of theories, all of them decidable in ⌣ c>0 ATIME (2 cn , n) , for which the subclasses with bounded quantifier alternations still have alternating exponential time complexity. In particular this yields non-deterministic exponential time lower bounds for very simple prefix classes (with 2 or 3 alternations). Theories with such behaviour are the theory of Boolean algebras, the theory of polynomial rings over finite fields, the theory of idempotent rings, the theory of finite sets with inclusion, the theory of semilattices, the theory of Stone algebras, the theory of distributive p -algebras in the Lee-class B n , and the theories of natural numbers with divisibility or coprimeness.

Pictures of functions and their acceptability by automata

Pictures of functions of one variable in the natural number theory are introduced as patterns over {0,1} in the first quadrant. It is proved that the set of pictures of all computable functions of one variable is obtained as the bottom planes of three-dimensional arrays accepted by some nondeterministic automaton on ω3-tapes. To our surprise, it is also shown that a set of pictures of functions containing a noncomputable function is deterministically obtained in a similar way. Further, it is proved that the set of initial segments of pictures of all computable functions and/or the noncomputable function is accepted by a deterministic Turing rectangular array acceptor.

Subclasses of presburger arithmetic and the polynomial-time hierarchy

We investigate the complexity of subclasses of Presburger arithmetic, i.e., the first-order theory of natural numbers with addition. The subclasses are defined by restricting the quantifier prefix to finite lists Q 1… Q s . For all m⩾ 2 we find formula classes, defined by prefixes with m+1 alternations and m+5 quantifiers, which are Σ p m - respectively Π p m -complete. For m=1, the class of ∃∀∀-formulas is shown to be NP-complete. For m=0 and for all natural numbers t, the class of ∃ t -formulas is known to be in P. Thus we have a nice characterisation of the polynomial-time hierarchy by classes of Presburger formulas. Finally, the NP-completeness of the ∃∀∀-class is used to prove that for certain formulas there exist no equivalent quantifier-free formulas of polynomial length.

Exactly Solvable IRF Models. IV. Generalized Rogers-Remanujan Identities and a Solvable Hierarchy

We propose a k -state IRF model ( k ≧2). The hard core conditions on these models are directly related to the generalized Rogers-Ramanujan identities in the partition theory of natural numbers. We predict that it is exactly solvable for all k . The conjecture holds for k =2, 3 and 4.

Decidability of the universal theory of natural numbers with addition and divisibility

The class of all quantifier-free formulas constructed from atomic formulas of the form (x+y= z),(x=1), and (x¦y) is considered, where the predicate symbol “¦” is interpreted as the divisibility relation on nonnegative integers. The decidability isproved of the set of all formulas of this form which are true for at least one choice of values for the variables. This result is equivalent to the decidability of the universal theory of natural numbers with addition and divisibility.

A normalization theorem in formal theories of natural numbers

A normalization theorem in formal theories of natural numbers

A finite arithmetic

In this paper I shall argue that the presumption of infinitude may be excised from the area of mathematics known as natural number theory with no substantial loss. Except for a few concluding remarks, I shall restrict my concern in here arguing the thesis to the business of constructing and developing a first-order axiomatic system for arithmetic (called ‘FA’ for finite arithmetic) that contains no theorem to the effect that there are infinitely many numbers.The paper will consist of five parts. Part I characterizes the underlying logic of FA. In part II ordering of natural numbers is developed from a restricted set of axioms, induction schemata are proved and a way of expressing finitude presented. A full set of axioms are used in part III to prove working theorems on comparison of size. In part IV an ordinal expression is defined and characteristic theorems proved. Theorems for addition and multiplication are derived in part V from definitions in terms of the ordinal expression of part IV. The crucial final constructions of part V present a new method of replacing recursive characterizations by strict definitions.In view of our resolution not to assume that there are infinitely many numbers, we shall have to deal with the situation where singular arithmetic terms of FA may fail to refer. For I know of no acceptable and systematic way of avoiding such situations. As a further result, singular-term instances of universal generalizations of FA are not to be inferred directly from the generalizations themselves. Nevertheless, (i) (x)(y)(x + y = y + x), for example, and all its instances are provable in FA.

An Investigation on the Logical Structure of Mathematics (VIII):: Consistency of the Natural-Number Theory T1(N)

The consistency of the natural number theory was proved, as is well known, by G. Gentzen in 1935 for the first time in such generality that the mathematical induction can be consistently used for any arbitrary predicate of natural numbers, which is well-formed in his system so that every quantifier ranges over all natural numbers.