The Arithmetization of Syntax and the New Paradoxes of Self-Reference

T. Parent
Virginia Tech
In this paper, I recreate a paradox from my earlier work (“Paradox with just self-reference”) albeit entirely within the language of arithmetic. (In lieu of self-referential expressions, Gödel numbering is used to generate the paradox.) This can suggest that Robinson arithmetic and its extensions are unsound; however, I claim instead that the metalanguage may be to blame, owing to arithmetization of syntax. If so, then the moral would be to restrict arithmetization in the metalanguage somehow, rather than distrust the arithmetical theory in the object language.
Keywords Gödel, Kurt  Paradox  Gödel numbering  Axiomatic arithmetic
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