Online research in philosophy

A new second-order axiomatization of arithmetic, with Frege's definition of successor replaced, is presented and compared to other systems in the field of Frege Arithmetic. The key in proving the Peano Axioms turns out to be a proposition about infinity, which a reduced subset of the axioms proves.

We consider extensions of Peano arithmetic suitable for doing some of nonstandard analysis, in which there is a predicate N(x) for an elementary initial segment, along with axiom schemes approximating ω 1 -saturation. We prove that such systems have the same proof-theoretic strength as their natural analogues in second order arithmetic. We close by presenting an even stronger extension of Peano arithmetic, which is equivalent to ZF for arithmetic statements.

Herein is presented a natural first-order arithmetic system which can prove its own consistency, both in the weaker Godelian sense using traditional Godel numbering and, more importantly, in a more robust and direct sense; yet it is strong enough to prove many arithmetic theorems, including the Euclidean Algorithm, Quadratic Reciprocity, and Bertrand’s Postulate.

Using an axiomatization of second-order arithmetic (essentially second-order Peano Arithmetic without the Successor Axiom), arithmetic's basic operations are defined and its fundamental laws, up to unique prime factorization, are proven. Two manners of expressing a system's consistency are presented - the "Godel" consistency, where a wff is represented by a natural number, and the "real" consistency, where a wff is represented as a second-order sequence, which is a stronger notion. It is shown that the system can prove at least its Godel consistency and that closely allied systems can prove their real consistency.

Second-order Peano Arithmetic minus the Successor Axiom is developed from first principles through Quadratic Reciprocity and a proof of self-consistency. This paper combines 4 other papers of the author in a self-contained exposition.

The Successor Axiom asserts that every number has a successor, or in other words, that the number series goes on and on ad infinitum. The present work investigates a particular subsystem of Frege Arithmetic, called F, which turns out to be equivalent to second-order Peano Arithmetic minus the Successor Axiom, and shows how this system can develop arithmetic up through Gauss' Quadratic Reciprocity Law. It then goes on to represent questions of provability in F, and shows that F can prove its own consistency and indeed the consistency of stronger systems. So, arithmetic without the Successor Axiom has an exceptional combination of three chracteristics: it is natural, it is strong, and it proves its own, as well as stronger systems’, consistency.

The system called F is essentially a sub-theory of Frege Arithmetic without the ad infinitum assumption that there is always a next number. In a series of papers (Systems for a Foundation of Arithmetic, True” Arithmetic Can Prove Its Own Consistency and Proving Quadratic Reciprocity) it was shown that F proves a large number of basic arithmetic truths, such as the Euclidean Algorithm, Unique Prime Factorization (i.e. the Fundamental Law of Arithmetic), and Quadratic Reciprocity, indeed a sizable amount of arithmetic. In particular, F proves some (but not all) of the Peano Axioms; that is, F proves the axioms of a sub-theory - call it FPA - of second-order Peano-Arithmetic. This short technical note will demonstrate that the converse also holds, in the following sense. F has the same language as second-order Peano Arithmetic except that, in addition, it has a two-place predicate symbol “Μ”. Then it is possible to provide a definition, indeed a reasonable definition, for “Μ” such that FPA proves all the axioms of F. So F and FPA effectively have the same proof-theoretic strength. In particular FPA, which lacks the Successor Axiom stating that every natural number has a successor, is able to prove the Euclidean Algorithm, Unique Prime Factorization, and Quadratic Reciprocity, indeed (again) a sizable amount of arithmetic.

The system of arithmetic considered in Consistency, which is essentially second-order Peano Arithmetic without the Successor Axiom, is used to prove more theorems of arithmetic, up to Quadratic Reciprocity.

In a short, technical note, the system of arithmetic, F, introduced in Systems for a Foundation of Arithmetic and "True" Arithmetic Can Prove Its Own Consistency and Proving Quadratic Reciprocity, is demonstrated to be equivalent to a sub-theory of Peano Arithmetic; the sub-theory is missing, most notably, the Successor Axiom.

A general method of interpreting weak higher-type theories of nonstandard arithmetic in their standard counterparts is presented. In particular, this provides natural nonstandard conservative extensions of primitive recursive arithmetic, elementary recursive arithmetic, and polynomial-time computable arithmetic. A means of formalizing basic real analysis in such theories is sketched.