Re: Python from Wise Guy's Viewpoint

From: Aatu Koskensilta (aatu.koskensilta_at_xortec.fi)
Date: 10/31/03 

Date: Fri, 31 Oct 2003 11:51:35 +0200

Ray Blaak wrote:

> My question is this: why is it the case that "there must be true sentences
> which are not provable"? Given that Tarski himself says "there exists a pair
> of contradictory sentences neither of which is provable", couldn't it be the
> case that the most you can say is "I don't know", i.e., I can't prove things
> either way?

This is possible, but not without giving up the assumption that the
language is closed under contradictory negation (and thus classical).
The problem with the answer "I don't know" is that we can consider a
sentence - known as the strengthened liar : "I don't know this sentence
to be true". I don't know it to be true, hence it's true, and hence I
*do* know that it's true.

There's a huge literature on this sort of problems, and various proposed
ways out.

> That is, I cannot decide on the sentence's truth, so for practical purposes I
> simply give up.

This makes you embrace a queer notion of truth: truth is simply being
known by you.

> In particular, Gödel sentences don't seem to have any truth meaning: any
> attempt to evaluate them give rises to the cyclical spinning of true then
> false then true..., i.e. an infinite loop in practical terms.

No they don't. You are still confusing provability in a particular
formal system with truth. For example, the Gödel statement for PA says
of itself that it's not provable, which is equivalent to the consistency
of PA. We know that PA is consistent, since it's true in the natural
numbers, and hence the sentence is true. There is no problem with the
liar when we are dealing with arithmetic sentences, simply because the
liar can't be formulated as an arithmetical statement.

Let's try: "this sentence is not provable in PA". Ok, assume it's not
provable in PA... what now? Where does the infinite loop come now?
Assume it's provable in PA. Ok, then PA is inconsistent (which it
isn't)... what now? The infinite loop you refer to simply does not occur. 

-- 
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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