On the arithmetization of syntax

Abstract

It is generally accepted that the incompleteness of first-order number theory (PA) is established by an application of Godel's proof. This paper shows that the arithmetization of the syntax of PA implies that the hypothesised class of PA theorems is not well defined. This is a consequence of the fact that the theoremhood (or non-theoremhood) of a PA formula is implied if the existence of a Godel number of a proof of the formula may be formally proved (or disproved respectively), using the systems own axioms and inference rules. Hence it is a PA theorem that the the negation of a Godel sentence for PA (Not G) implies G itself, i.e. ((Not G) => G) is a PA theorem; from which, both the theoremhood and non-theoremhood of G may be established. The conclusion is taken as evidence of a failure of the devices relied upon for the avoidance of paradox in metamathematical definitions of a proof, a formal theory etc. The main proposition is derived using only assumptions supported by proofs already accepted in the existing literature.