Re: Liar's Paradox in Godel's Theorem (newbie question)


David C. Ullrich wrote:
On 3 May 2006 17:33:11 -0700, "Charlie-Boo" <shymathguy@xxxxxxxxx>
wrote:


David C. Ullrich wrote:
On 2 May 2006 21:02:44 -0700, "Charlie-Boo" <shymathguy@xxxxxxxxx>
wrote:

David C. Ullrich wrote:
On 30 Apr 2006 16:47:01 -0700, "Charlie-Boo" <shymathguy@xxxxxxxxx>
wrote:

Aatu Koskensilta, Charlie-Boo wrote:

You don't believe in correcting the record, now that you see the close
relationship between Liar and Godel?

There's nothing to correct. It's not particularly illuminating to know
that "this sentence is not true" is not equivalent to "this sentence is
not provable in PA".

It proves Godel's Theorem.

Right. Exactly how does this proof go?

As I said above, "Do you see how to derive Godel's Theorem based on
soundness given that truth and provability do not coincide?"

Uh, that doesn't answer the question. Given that truth and
provability do not coincide, now how do we prove Godel's theorem?

There must be a sentence that is not true and provable (the logic in
not sound), or true and not provable (an implication if the logic is
sound, which is the theorem.) I thought that was too trivial to
mention.

Probably we should back up. For starters, why don't you
give a precise statement of the theorem of Godel you're
referring to?

If a logic is sound and can express the set of Godel numbers of the
unprovable sentences (any non-representable set), then there is a
sentence that is not provable and not refutable.

(Hint regarding why this is nonsense: There are some quantifiers
in the statement of the theorem: For _any_ theory T satisfying
certain conditions, ...)

Yes, the ability to represent the above.

You seem to be missing the point.

No, you missed a trivial proof.

Given that truth and provability
are not the same, how does it follow that for _any_ theory T
satisfying these hypotheses there exists a sentence S such
that T proves neither S nor ~S?

Given that truth and provability are not the same, soundness implies
incompleteness by the above trivial proof.

************************

David C. Ullrich

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