[FOM] extensions of PA that prove "false" statements

[Aatu Koskensilta]

On May 18, 2006, at 10:53 PM, praatika at mappi.helsinki.fi wrote:

> Let Cons(PA) be the arithmetical formula expressing the statement that 
> PA
> is consistent. By Gödel's second theorem, neither Cons(PA) nor its
> negation is provable in PA.

This of course depends on Sigma-soundness (1-consistency) of PA. In 
full generality Gödel's second incompleteness theorem only tells us 
that a consistent T (satisfying certain criteria) does not prove its 
consistency; if T is Sigma-unsound it might well incorrectly prove that 
it's inconsistent. All of this explained in Torkel's book.

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

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