Re: Deep Thoughts # 17: Liar Paradox is a Formal Metamathematical Theorem

From: Daryl McCullough (daryl_at_atc-nycorp.com)
Date: 12/26/04 

Date: 26 Dec 2004 12:28:24 -0800

Charlie-Boo says...

>We already know what provable means because Smullyan just defined it to
>be truth in the first phrase.

No, he didn't. The quote that you give from Smullyan does not
say that at all. He says "thus" the provable sentences and the
true sentences coincide for his system. "Thus" implies that he
is drawing a *conclusion* rather than making a definition. It
*follows* from the *usual* definition of "provable" that for a
complete theory, the true sentences and the provable sentences
coincide.

>So, instead of saying, "Smullyan defines
>provability to be truth.", we should instead say, "Smullyan defines
>provability to be truth with the help of coauthors." Is that better?

It's better to say that Smullyan defines the axioms of his system to
be all true sentences. Therefore, it follows from the usual definition
of "provable" that the provable sentences and the true sentences
coincide for his system.

At no point is Smullyan defining provability in terms of truth.

>Actually, (A) "the first-order system whose axioms are all the
>correct formulas" does not imply (B) "the provable formulas of N are
>nothing more than the axioms of N."

Yes, it does.

>A is about the axioms and B is about the relationship between the
>derivation rules and the axioms. Regardless of what A says, B can be
>true or false depending on the derivation rules.

The derivation rules are fixed by the fact that we are dealing
with first-order logic. (Or rather, I should say that the logical
consequence relation is fixed by the fact that we are dealing
with first-order logic.)

>A => B here only because Smullyan is also (must be) assuming that
>(B') the derivation rules produce nothing outside of the axioms.'

That's not an assumption, that is an easily proved *conclusion*.

>In fact, given assumption B', B is merely
>equivalent to this assumption and statement A plays no role in the
>truth of B at all.

Well, that's completely wrong, but I think that has already
been pointed out before. 

--
Daryl McCullough
Ithaca, NY

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