Re: Godel's comments about the "true reason" for incompleteness

On 2008-03-22, in sci.logic, R. Srinivasan wrote:
Now consider the following sentence of T:

(Fot all P) (For all Q) (P&~P)->Q (*)

where P and Q range over sentences in the language of PA.

(*) is a sentence expressible in the language of T.

(*) is not a sentence in a theory directly formalising the syntax of
PA. It is a formula in quantified propositional logic. Certainly we
can produce a translation from quantified propositional logic into PA,
that is, a mapping S |--> S' such that if S is provable in quantified
propositional logic S' is provable in PA, but not in the another
direction. This translation is also something that we make no use of
in the proof of the incompleteness theorem.

Thus we have the strange situation in which two supposedly equivalent
theories PA and T are not equivalent after all. What do you think
about this?

Your argument is confused.

OK, but you do need to quantify over the formal sentences.

Sure, but only as strings of symbols.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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