Zhang Hong recently asked "is there a paradox lurking in Godel's 1931 incompleteness proof" (paraphrase)? This can be answered in two ways: First, by proving quite generally that there are no paradoxes in mathematical logic. Or, second, by actually producing a paradoxical result within mathematical logic.
The following is putatively an answer of the second kind, to Hong's question. I'm posting here to ask, "is the following a paradox, or just a mistake?"
- Let #Q# be the Godel number of the expression, Q.
- Let @r@ be the expression corresponding to the Godel number, r.
- Now let n = #@n@ --> P#, with P being arbitrary.
- Then P.
(Note that Q = @#Q#@ and r = #@r@#, for all expressions, Q, and all Godel numbers, r.)
The proof from 3 to 4 goes this way: Suppose @n@. Then by applying @..@ quotes to both sides of 3, it follows that @n@ --> P. So it follows that P (by modus ponens). Suppose ~@n@, contrariwise. Then @n@ --> P is false. So the antecedent is true and the conclusion is false. Thus it follows that @n@ (since this is the antecedent). And it follows from this that @n@ --> P. So we have it again (by modus ponens) that P. So, whether @n@ is true or false, it yeilds a proof that P, even though P could be anything.
Edit: It was suggested that I add a link to Zhang Hong's question. So here it is: Hong's question.
