Axiomatizing Mathematical Theories: Multiplication

Fundamenta Informaticae

The multiplicative theory of a set of numbers (which could be natural, integer, rational, real or complex numbers) is the first-order theory of the structure of that set with (solely) the multiplication operation (that set is taken to be multiplicative, i.e., closed under multiplication). In this paper we study the multiplicative theories of the complex, real and (positive) rational numbers. These theories (and also the multiplicative theories of natural and integer numbers) are known to be decidable (i.e., there exists an algorithm that decides whether a given sentence is derivable form the theory); here we present explicit axiomatizations for them and show that they are not finitely axiomatizable. For each of these sets (of complex, real and [positive] rational numbers) a language, including the multiplication operation, is introduced in a way that it allows quantifier elimination (for the theory of that set).

Mathematics of the USSR – Izvestia. 15:2, 1980

In this paper the problems of expressibility and decidability are studied for elementary theories obtained by extending the arithmetic of order and the arithmetic of addition of natural numbers. Results are obtained on the decidability and undecidability of elementary theories of concrete structures of the form ⟨N;+,P⟩, where P is a fixed monadic predicate, as well as results on the class of sets definable in the theory T⟨N;+,λx,∃y(x=dy)⟩.

Axioms

The systems of arithmetic discussed in this work are non-elementary theories. In this paper, natural numbers are characterized axiomatically in two different ways. We begin by recalling the classical set P of axioms of Peano’s arithmetic of natural numbers proposed in 1889 (including such primitive notions as: set of natural numbers, zero, successor of natural number) and compare it with the set W of axioms of this arithmetic (including the primitive notions like: set of natural numbers and relation of inequality) proposed by Witold Wilkosz, a Polish logician, philosopher and mathematician, in 1932. The axioms W are those of ordered sets without largest element, in which every non-empty set has a least element, and every set bounded from above has a greatest element. We show that P and W are equivalent and also that the systems of arithmetic based on W or on P, are categorical and consistent. There follows a set of intuitive axioms PI of integers arithmetic, modelled on P and propos...

Soft Computing

The ordered structures of natural, integer, rational and real numbers are studied here. It is known that the theories of these numbers in the language of order are decidable and finitely axiomatizable. Also, their theories in the language of order and addition are decidable and infinitely axiomatizable. For the language of order and multiplication, it is known that the theories of N and Z are not decidable (and so not axiomatizable by any computably enumerable set of sentences). By Tarski's theorem, the multiplicative ordered structure of R is decidable also; here we prove this result directly and present an axiomatization. The structure of Q in the language of order and multiplication seems to be missing in the literature; here we show the decidability of its theory by the technique of quantifier elimination and after presenting an infinite axiomatization for this structure we prove that it is not finitely axiomatizable.

2007

2020

The ordered structures of natural, integer, rational and real numbers are studied in this thesis. The theories of these numbers in the language of order are decidable and finitely axiomatizable. Also, their theories in the language of order and addition are decidable and infinitely axiomatizable. For the language of order and multiplication, it is known that the theories of $\mathbb{N}$ and $\mathbb{Z}$ are not decidable (and so not axiomatizable by any computably enumerable set of sentences). By Tarski's theorem, the multiplicative ordered structure of $\mathbb{R}$ is decidable also. In this thesis we prove this result directly by quantifier elimination and present an explicit infinite axiomatization. The structure of $\mathbb{Q}$ in the language of order and multiplication seems to be missing in the literature. We show the decidability of its theory by the technique of quantifier elimination and after presenting an infinite axiomatization for this structure, we prove that it i...

One of the richest and most salient applications of a non-classical logic is the matter of how mathematics operates within its province. Historically, this is most evident in the case of intuitionism, insofar as the intuitionistic standpoints with respect to deduction and mathematical practice are tightly bound together. Yet even in the case of Robert Meyer's relevant arithmetic R#, that a robust and compelling theory of arithmetic can be erected on relevant foundations speaks to the maturity of relevant logics. Accordingly, as connexive logic matures as a field, the topography of mathematics against a connexive backdrop becomes more and more compelling. The contraclassicality of connexive logics entails that the development of connexive mathematics will be more complex---and, arguably, more interesting---than intuitionistic or relevant accounts. For example, although formally undecidable sentences in classical Peano arithmetic remain independent of its intuitionistic and relevant counterparts, there exist undecidable sentences of classical arithmetic that will become decidable modulo any reasonable connexive arithmetic. In, e.g., Peano arithmetic, the Gödel sentence G is undecidable. Classically, this entails that the sentence ~(G->~G) is likewise undecidable. Of course, in a connexive logic L and connexive arithmetic L#, L# will prove ~(G->~G), witnessing that some classically undecidable statements in number theory become decidable connexively. Although this example is extremely simple, it demonstrates that there are many subtle questions that uniquely arise in a connexive mathematics. In this paper, I wish to make a few comments on how mathematics---in particular, arithmetic---must behave if formulated connexively. We will first consider some relevant historical and philosophical topics, such as Łukasiewicz' number-theoretic argument against Aristotle's Thesis, before taking a foray into the formalization of modest subsystems of arithmetic in Richard Angell's PA1 and PA2, observing some of the pathologies that will greet arithmetic in these settings.

arXiv (Cornell University), 2017

The ordered structures of natural, integer, rational and real numbers are studied here. It is known that the theories of these numbers in the language of order are decidable and finitely axiomatizable. Also, their theories in the language of order and addition are decidable and infinitely axiomatizable. For the language of order and multiplication, it is known that the theories of N and Z are not decidable (and so not axiomatizable by any computably enumerable set of sentences). By Tarski's theorem, the multiplicative ordered structure of R is decidable also; here we prove this result directly and present an axiomatization. The structure of Q in the language of order and multiplication seems to be missing in the literature; here we show the decidability of its theory by the technique of quantifier elimination and after presenting an infinite axiomatization for this structure we prove that it is not finitely axiomatizable.

The Journal of Symbolic Logic, 2005

We investigate theories of initial segments of the standard models for arithmetics. It is easy to see that if the ordering relation is definable in the standard model then the decidability results can be transferred from the infinite model into the finite models. On the contrary we show that the Σ2-theory of multiplication is undecidable in finite models. We show that this result is optimal by proving that the Σ1-theory of multiplication and order is decidable in finite models as well as in the standard model. We show also that the exponentiation function is definable in finite models by a formula of arithmetic with multiplication and that one can define in finite models the arithmetic of addition and multiplication with the concatenation operation.

2017

The ordered structures of natural, integer, rational and real numbers are studied here. It is known that the theories of these numbers in the language of order are decidable and finitely axiomatizable. Also, their theories in the language of order and addition are decidable and infinitely axiomatizable. For the language of order and multiplication, it is known that the theories of N and Z are not decidable (and so not axiomatizable by any computably enumerable set of sentences). By Tarski's theorem, the multiplicative ordered structure of R is decidable also; here we prove this result directly and present an axiomatization. The structure of Q in the language of order and multiplication seems to be missing in the literature; here we show the decidability of its theory by the technique of quantifier elimination and after presenting an infinite axiomatization for this structure we prove that it is not finitely axiomatizable.