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


Authors
T. Parent
Virginia Tech
Abstract
In this paper, I recreate a paradox from my earlier work (“Paradox with just self-reference”) albeit entirely within the language of arithmetic. The paradox might suggest that Robinson arithmetic and its extensions are unsound; however, I claim instead that the metalanguage may be to blame, owing to the lack of restrictions on the 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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The Semantic Conception of Truth and the Foundations of Semantics.Alfred Tarski - 1943 - Philosophy and Phenomenological Research 4 (3):341-376.
The Completeness of the First-Order Functional Calculus.Leon Henkin - 1949 - Journal of Symbolic Logic 14 (3):159-166.

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