So far as a possible canonical model of PA is concerned, it can only be one of the two distinctly different—and hitherto unsuspected—constructive interpretations of the first order Peano Arithmetic PA over the natural numbers that are detailed in a paper  I had presented in Birmingham last July at the Symposium on Computational Philosophy, AISB/IACAP World Congress 2012-Alan Turing 2012.

(i) The first is isomorphic to the standard interpretation of PA over the natural numbers. Under this interpretation the satisfaction and truth of the atomic formulas of PA is defined in terms of algorithmic verifiability in such a way that, by Tarski’s definitions, the true compound sentences of PA under the interpretation are inductively defined as those PA-propositions that are algorithmically verifiable as true under the interpretation.

This interpretation does not offer a finitary interpretation of the Axiom Schema of Finite Induction, so it cannot be taken to define a model of PA from a finitary perspective.

(ii) The second is an algorithmic interpretation over the natural numbers. Under this interpretation the satisfaction and truth of the atomic formulas of PA is defined in terms of algorithmic computability in such a way that, by Tarski’s definitions, the true compound sentences of PA under the interpretation are inductively defined as those PA-propositions that are algorithmically computable as true under the interpretation.

This interpretation does offer a finitary interpretation of the Axiom Schema of Finite Induction, so it can be taken to define a model of PA from a finitary perspective, and could be considered a candidate for a canonical model of PA.

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

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