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arXiv · 1105.0321

On first-order arithmetic truth

Stephen Boyce
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Abstract

The standard interpretation of first-order number theory (PA), according to the generally accepted view, associates well-defined set-theoretic entities with each and every well-formed formula of this system. But this implies that the class of PA theorems is semantically defined by a class sign of PA itself, (E x_2) Pf(x_2, x_1), in the following sense: with b' the PA numeral for the number b, (E x_2) Pf(x_2, b') is true under the standard interpretation if and only if b is the Godel number of a PA theorem. From this however it is easily established, by a modification of Godel's proof, that the class of PA theorems, and hence the standard interpretation of PA itself, is not well defined after all.

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Stephen Boyce. 2011-05-02. On first-order arithmetic truth. https://arxiv.org/abs/1105.0321

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