Topic: GENERAL LOGIC
Diagonalization and Self-Reference
By Richard Heck
It is often said that diagonalization allows one to construct sentences that are self-referential. This paper investigates the sense in which that is true. I argue first that, in the standard language of arithmetic, in which we have only the symbols 0, S, +, and ?, truly self-referential sentences cannot be constructed. This is shown by considering sentences like The right-hand side of this biconditional is false iff its left-hand side is true. This sentence is intuitively inconsistent, but the sentence constructed by using diagonalization in the usual way is true and, in fact, provable in Q. This problem can be resolved by expanding the language to include function-symbols for all primitive recursive functions. It can also be resolved by proving a stronger form of the diagonal lemma that I call the structural diagonal lemma. At the end of the paper, it is suggested, however, that there are some contexts in which even these methods are insufficient.
Source: PH Online, Online Papers in Philosophy
Posted by Tony Marmo at 18:41 BST