Some consequences of interpreting the associated logic of the first-order Peano Arithmetic PA finitarily

TL;DR

This paper explores finitary interpretations of first-order Peano Arithmetic, showing that its axioms and rules are algorithmically computable and that a finitary interpretation can establish PA's consistency without relying on non-finitary methods.

Contribution

It introduces a finitary interpretation of PA based on algorithmic computability, challenging traditional non-finitary views of its standard interpretation.

Findings

Classical interpretations allow defining satisfaction and truth via algorithmic verifiability and computability.

PA axioms and inference rules are algorithmically computable and preserve satisfiability under the standard interpretation.

A finitary interpretation of PA demonstrates its categoricity and vacuous validity of Gödel's Theorem VI, asserting consistency without omega-consistency.

Abstract

We show that the classical interpretations of Tarski's inductive definitions actually allow us to define the satisfaction and truth of the quantified formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two essentially different ways: (a) in terms of algorithmic verifiabilty; and (b) in terms of algorithmic computability. We show that the classical Standard interpretation I_PA(N, Standard) of PA essentially defines the satisfaction and truth of the formulas of the first-order Peano Arithmetic PA in terms of algorithmic verifiability. It is accepted that this classical interpretation---in terms of algorithmic verifiabilty---cannot lay claim to be finitary; it does not lead to a finitary justification of the Axiom Schema of Finite Induction of PA from which we may conclude---in an intuitionistically unobjectionable manner---that PA is consistent. We now show that the PA-axioms---including the Axiom Schema of Finite Induction---are, however, algorithmically computable finitarily as satisfied / true under the Standard interpretation I_PA(N, Standard) of PA; and that the PA rules of inference do preserve algorithmically computable satisfiability / truth finitarily under the Standard interpretation I_PA(N, Standard). We conclude that the algorithmically computable PA-formulas can provide a finitary interpretation I_PA(N, Algorithmic) of PA from which we may classically conclude that PA is consistent in an intuitionistically unobjectionable manner. We define this interpretation, and show that if the associated logic is interpreted finitarily then (i) PA is categorical and (ii) Goedel's Theorem VI holds vacuously in PA since PA is consistent but not omega-consistent.