A History of Interactions between Logic and Number Theory

American Mathematical Monthly, 1991

Logic Journal of the IGPL, Volume 32, Issue 5, Pages 880-908, 2024

In this work, we aim at understanding incompleteness in an abstract way via metamathematical properties of formal theories. We systematically examine the relationships between the following twelve important metamathematical properties of arithmetical theories: Rosser, EI (effectively inseparable), RI (recursively inseparable), TP (Turing persistent), EHU (essentially hereditarily undecidable), EU (essentially undecidable), Creative, 0 (theories with Turing degree 0), REW (all RE sets are weakly representable), RFD (all recursive functions are definable), RSS (all recursive sets are strongly representable), RSW (all recursive sets are weakly representable). Given any two properties P and Q in the above list, we examine whether P implies Q.

arXiv: Logic, 2020

This paper is a survey on model theory of adeles and applications to model theory, algebra, and number theory. Sections 1-12 concern model theory of adeles and the results are joint works with Angus Macintyre. The topics covered include quantifier elimination in enriched Boolean algebras, quantifier elimination in restricted products and in adeles and adele spaces of algebraic varieties in natural languages, definable subsets of adeles and their measures, solution to a problem of Ax from 1968 on decidability of the rings $\mathbb{Z}/m\mathbb{Z}$ for all $m>1$, definable sets of minimal idempotents (or "primes of the number field" ) in the adeles, stability-theoretic notions of stable embedding and tree property of the second kind, elementary equivalence and isomorphism for adele rings, axioms for rings elementarily equivalent to restricted products and for the adeles, converse to Feferman-Vaught theorems, a language for adeles relevant for Hilbert symbols in number theo...

1979

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...

Studies in Logic and the Foundations of Mathematics, 1998

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

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.

Proceedings of The American Mathematical Society, 1988

Every countable model M of PA or ZFC, by a theorem of S. Simpson, has a "class" X which has the curious property: Every element of the expanded structure (M, X) is definable. Here we prove: THEOREM A. Every completion T of PA has a countable model M (indeed there are 2" many such M 's for each T) which is not pointwise definable and yet becomes pointwise definable upon adjoining any undefinable class X to M. THEOREM B. Let M 1= ZF + "V = HOD" be a well-founded model of any cardinality. There exists an undefinable class X such that the definable points of M and (M, X) coincide. THEOREM C. Let M t= PA or ZF +"V = HOD". There exists an undefinable class X such that the definable points of M and (M, X) coincide if one of the conditions below is satisfied. (A) The definable elements o/M are cofinal in M. (B) M is recursively saturated and cf (M) = uj.

2016

The theory of addition in the domains of natural (N), integer (Z), rational (Q), real (R) and complex (C) numbers is decidable, so is the theory of multiplication in all those domains. By Godel's Incompleteness Theorem the theory of addition and multiplication is undecidable in the domains of N, Z and Q, though Tarski proved that this theory is decidable in the domains of R and C. The theory of multiplication and order (x,<) behaves differently in the above mentioned domains of numbers. By a theorem of Robinson, addition is definable by multiplication and order in the domain of natural numbers, thus the theory (N;x,<) is undecidable. By a classical theorem in mathematical logic, addition is not definable in terms of multiplication and order in R. In this paper, we extend Robinson's theorem to the domain of integers (Z) by showing the definability of addition in (Z;x,<), this implies that (Z;x,<) is undecidable. We also show the decidability of (Q;x,<) by the m...