Models and Recursivity

Abstract

It is commonly held that the natural numbers sequence 0, 1, 2,... possesses a unique structure. Yet by a well known model theoretic argument, there exist non-standard models of the formal theory which is generally taken to axiomatize all of our practices and intentions pertaining to use of the term “natural number.” Despite the structural similarity of this argument to the influential set theoretic indeterminacy argument based on the downward L ̈owenheim-Skolem theorem, most theorists agree that the number theoretic version does not have skeptical consequences about the reference of “natural number” analogous to the ‘relativity’ Skolem claimed pertains to notions such as “uncountable” and “cardinal.” In this paper I argue that recent proposals by Shapiro, Lavine, McGee and Field which aim to distinguish the number and set theoretic indeterminacy arguments by locating extra-mathematical constraints on the interpretation of our number theoretic vocabulary are inadequate. I then suggest that if we consider the manner in which the natural numbers figure in our computational practices – e.g. in characterizing how we compute sums and products or determine prime factorizations – it follows that our application of computational terms such as “computable function” and “tractable computational problem” indirectly uniquely determines their structure.

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References found in this work

Reason, truth, and history.Hilary Putnam - 1981 - New York: Cambridge University Press.
Reason, Truth and History.Hilary Putnam - 1981 - New York: Cambridge University Press.
What numbers could not be.Paul Benacerraf - 1965 - Philosophical Review 74 (1):47-73.
On Computable Numbers, with an Application to the Entscheidungsproblem.Alan Turing - 1936 - Proceedings of the London Mathematical Society 42 (1):230-265.

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