[FOM] CH and mathematics

Bill Taylor W.Taylor at math.canterbury.ac.nz
Wed Jan 23 23:32:14 EST 2008

->> > Is Feferman's question in effect an expression of doubt about
->> > the meaningfulness of CH? Would you say that the same would have been
->> > true of Fermat's last theorem would have been proved independent of
->> > the axioms of arithmetic?
->> These are in no way parallel.  FLT is arithmetical, so is subject
->> to the usual straightforward quantifier-extension in meaning as explicated
->> from Hilbert to Tarski.  No such remarks apply to set-theoretical
->> results, as Feferman was at pains to make clear in his quote, I feel.
->I'm not sure what the "quantifier-extension in meaning" is. 

Sorry for being obscure; I should have expanded, originally.

I just intend the usual and "obvious" meaning:- that delta_0 statements
have the meaning that they can be checked in finite time; pi_1 statements
are false when a certain TM (that checks the statement) eventually halts;
sigma_0 ones are true likewise; pi_2 are true when a certain machine
that outputs other machines' codes, halts on all the sub-machines, and
thus the original continues on ever higher code numbers; and so on.

Thus, every ARITHMETICAL statement can be given a clear-cut meaning,
whether or not we can ever determine (by proof or otherwise) if it is
true or false.

However SET_THEORETIC statements cannot all be given such a simple meaning.
That is why FLT is not comparable with CH in this way.

-> I think Alex makes 2 very valid points in his above quote:
-> 1) Is "inherently vague" the same thing as "meaningless"? 

I read Feferman as saying, (and I agree with him), that for SET_THEORETIC
statements, that is so.  But not for arithmetical ones.

-> Presumably "meaningless" would imply "uninteresting".

Oh no!   There is no warrant for this at all!

They may be interesting for all sorts of reasons.
Among these would be that they might imply that certain pi_1 or sigma_1
ARITHMETICAL statements were undecidable within PA.  This would automatically
entail that those were true or false repsectively.

Hilbert pretty much said this, in not quite so many words, in his
article where he spoke of systems extending basic arithmetic.
(Sorry I forget the exact quote or source right now.)

-> 2) There is (I think) the remaining question of whether the current proof of
-> FLT can be formalized within PA.If it cannot, does that mean Wiles' proof of
-> FLT is inherently vague? 

I guess that depends on your view of consis(ZFC).  If you are happy that
this is true, then you will be happy that FLT is unfalsifiable in PA,
and thus true.   In any event, FLT itSELF is still meaningful, either way,
being arithmetical.

Bill Taylor

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