Arithmetic on semigroups

The Australasian Journal of Logic, 2021

We assess Meyer’s formalization of arithmetic in his [21], based on the strong relevant logic R and compare this with arithmetic based on a suitable logic of meaning containment, which was developed in Brady [7]. We argue in favour of the latter as it better captures the key logical concepts of meaning and truth in arithmetic. We also contrast the two approaches to classical recapture, again favouring our approach in [7]. We then consider our previous development of Peano arithmetic including primitive recursive functions, finally extending this work to that of general recursion.

Notre Dame Journal of Formal Logic, 2015

In this paper we study the interpretations of a weak arithmetic, like Buss's theory S 1 2 , in a given theory U. We call these interpretations the arithmetics of U. We develop the basics of the structure of the arithmetics of U. We study the provability logic(s) of U from the standpoint of the framework of the arithmetics of U. Finally, we provide a deeper study of the arithmetics of a finitely axiomatized sequential theory.

Grazer Philosophische Studien, 1998

2011

1996

This paper is a survey of recent results in the theory of nite semigroups using syntactic techniques. This involves manipulating pseudoidentities and the study of relatively free pro nite semigroups, whose structure encodes algebraic and combinatorial properties common to the members of a pseudovariety of semigroups. These techniques are illustrated with applications in the study of semigroups of order-preserving mappings, R-trivial semigroups, and semidirect products.

Notre Dame Journal of Formal Logic, 1998

After a summary of earlier work it is shown that elementary or Kalmar arithmetic can be interpreted within the system of Russell's Principia Mathematica with the axiom of infinity but without the axiom of reducibility. 1 Historical introduction After discovering the inconsistency in Frege's Grundgesetze der Arithmetik, Russell proposed two changes: first, dropping the assumption that to every higher-order entity there corresponds a first-order entity; and second, restricting the assumptions on the existence of higher-order entities, so that instead of a simple hierarchy of first-order, second-order, third-order, and so on, one has a ramified hierarchy in which each order is subdivided into various types in such a way that a condition involving quantification over all entities of one type is never assumed to determine another entity of the same type, but only of a higher type. But Russell found that with these two changes he could not derive classical mathematics, so in Principia Mathematica he partially compensated for the first change by assuming the axiom of infinity and, for all mathematical purposes, wholly undid the second change by assuming his axiom of reducibility. The predicativist tradition from Weyl [21] to Feferman [2] and beyond accepts infinity but rejects reducibility and is willing to give up parts of classical mathematics. However, predicativists have been unable to derive classical arithmetic and unwilling to give it up and so have simply assumed it as axiomatic. This assumption has its defenders, as with Feferman and Hellman [3], and also its detractors, as with C. Parsons [15]. It is, therefore, of some philosophical as well as historical interest to ask how large a fragment of classical arithmetic can be developed within the Russellian system of Principia Mathematica with infinity but without reducibility. Now many subsystems of classical or Peano arithmetic have been recognized since the work of Skolem [18], Kalmar [9], Grzegorczyk [4], and other pioneers. Among these the most studied have been the subprimitive or Grzegorczyk arithmetics n. These agree in allowing definitions by primitive recursion, but only when the function F being defined recursively is bounded by some function already given; or

Anais do II Workshop Brasileiro de Lógica (WBL 2021), 2021

This paper presents the (second-order) theory of hereditarily finite sets according to the usual pattern adopted in the presentation of the (second-order) theory of natural numbers. To this purpose, we consider three primitive concepts, together with four axioms, which are analogous to the usual Peano axioms. From them, we prove a homomorphism theorem, its converse, categoricity, and a kind of (semantical) completeness.

Theoretical Computer Science, 2004

In this paper, we introduce and study some syntactical fragments of monadic second-order and ÿrst-order (PEANO) arithmetic which we will prove the connection to famous complexity classes. Starting from descriptive complexity results, and giving an e ective method for translating formulas between di erent logical structures representing encodings of integers, we give some new arithmetical characterizations of NP, PH, NL, and P.

1997

2007