[FOM] Godel, Wittgenstein etc.
William Tait
wwtx at earthlink.net
Wed May 7 10:26:05 EDT 2003
On Tuesday, May 6, 2003, at 04:02 AM, praatika at mappi.helsinki.fi wrote:
> Several postings here seem to suggest the the *truth* of G for PA
> depends
> on its provability in PA^2 or ZFC or whatsoever.
Perhaps I am included in this indictment. I would note, however, that I
did not say that the truth of G _depends_ on its provability in PA^2. I
said that the only warrant we have for asserting G (which is the same
thing as asserting its truth) is having a proof of it. This should not
be understood to define truth = provability.
> I think that this is
> unnecessary and misleading. I've been trying to remind people that its
> truth is much more elementary issue, for it is entailed by the
> assumption
> that PA is consistent.
That is very true: but we are also only warranted in asserting the
mathematical proposition expressing the consistency of PA on the basis
of a proof. I chose PA^2 because it is the natural extension of PA in
which we can prove this truth (using a formal truth definition).
> The issue becomes important when one considers e.g.
> ZFC (or whatever is the strongest system one can take seriously;
> perhaps
> ZFC + the existence of some extremely large cardinals, or whatever).
> Clearly one cannot then appeal to the provability in ZFC in order to
> argue
> that G for ZFC is true.
Yes, this was one of Goedel's arguments for the essential
incompleteness of mathematics. To formalize the intuitive grounds upon
which we see that G(ZF+whatever) is true, we need ZF+ whatever +
something more.
.
> But we have exactly as much warrant to believe in
> the truth of G(ZCF) as we have warrant to believe in the consistency of
> ZFC; and similarly for other theories... At some point, this is not a
> matter of mathematical proof, but a matter of the degree of confidence
> we
> have on the consistency of a given theory (and ti may vary form theory
> to
> theory).
We are obviously using the term warrant in different ways. Degrees of
confidence certainly play a role in what axioms we are willing to
accept. But questions of truth in mathematics, I would argue, become
precise only on the basis of having accepted axioms.
Of course, there is another, non-semantical, use of the term
``true''---true to the conception we have in mind. In this sense of
`true', truth can indeed be a guide to the choice of axioms. (For the
conception of Euclidean space, the axiom of parallels is true.)
> Also, I think that the appeal to truth definitions is both unnecessary
> and
> misleading. As I noted before, it is often possible to give an adequate
> truth definition in a conservative extension, e.g. in NBG for ZFC
> (and in
> ACA_0 for PA); but such an extension does *not* prove the truth of
> Godel
> sentence. Also, the real point seems to be that (for a given system F):
>
> Cons(F) => (-Prov([G]) & G).
>
> It adds little to say that:
>
> Cons(F) => (-Prov([G]) & True([G]).
>
> A minimal theory of truth (i.e. just adding T-sentences) suffices for
> that
> move; but it is a conservative extension too. Arguably the notion of
> truth
> does no real work here.
Don't focus on formal truth definitions: focus on the reason we believe
that G is true. Namely, we can prove it. I would say that the most
natural proof is via formal truth definition in PA^2; but maybe that's
wrong. But, anyway, using truth definitions (call it something else if
the term `truth' is bothering you) is only a tool in the proof. The
connotations associated with the term `true' are not appealed to in the
proof.
In the proof of G in question, the formal truth definition does a good
deal of work. Referring to weak `theories of truth' is entirely
irrelevant.
With kind regards,
Bill Tait
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