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

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

2011-08-20

arxiv.org/abs/1108.4598v1

Mathematics

General Mathematics

32pages. an updated version of this manuscript is accessible at http://alixcomsi.com/30_Finitary_Logic_PA_Update.pdf . arXiv a

Scientific paper

We show that Tarski's inductive definitions actually allow us to define the satisfaction and truth of the formulas of the first-order Peano Arithmetic PA constructively 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 standard interpretation of PA defines the satisfaction and truth of the formulas of the first-order Peano Arithmetic PA constructively in terms of algorithmic verifiability. It is accepted that this interpretation 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. However, we now show that the PA-axioms---including the Axiom Schema of (finite) Induction---are algorithmically computable as satisfied / true under the standard interpretation of PA, and that the PA rules of inference preserve algorithmically computable satisfiability / truth under the standard interpretation. We conclude that the algorithmically computable PA-formulas can provide a finitary interpretation of PA from which we may conclude that PA is consistent in an intuitionistically unobjectionable manner. We define such an 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. This reflects the fact that PA is omega-consistent if, and only if, Aristotle's particularisation is presumed to always hold under any interpretation of the associated logic; and that the standard interpretation of PA is a model of PA if, and only if, PA is omega-consistent.

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