FOM: a question re. completeness

[michael Detlefsen]

Vladimir Konvei finds my formulation of consistency-completeness
meaningless. I don't see why. Indeed, I think we use exactly the same type
of counterfactual to describe the unprovability of the Goedel sentence: T
would be inconsistent if G were provable in it. Is this meaningless?

VK then goes on to say what he thinks I SHOULD mean ... to wit:

T is consistency-complete iff for every sentence s of the
language of T that is not provable in T, the following is
a theorem of T:
if Prov_T(s) then \neg Consis s

Following that, he says that he's not sure this is meaningful either. Boy,
VK, there doesn't seem to be much meaning in your world!!

I don't think you said quite what you meant (either that, or I don't have
idea why you think your definition would be a reformulation of what I had
in mind. I think you meant to say:

{*} T is consistency-complete iff for every sentence s of the
language of T that is not provable in T, the following is
a theorem of T:
if Prov_T(s) then \neg Consis T

(This is just like yours except where you had 'neg Consis s' I put 'neg
Consis T'.)

This latter, I think, is meaningful and yields a clear answer to (what some
have thought) is my question. Put 'neg ConT' in for 's' and apply Lob's
theorem. Then, since 'neg ConT' is not provable in T (for the usual T), you
know that your conditional is not provable in T.

I wrote my posting of this morning fully aware of this fact (i.e. the fact
that for ordinary T, the conditional ' Prov-T(neg ConT)-->neg ConT' is not
provable in T). Indeed, though thinking of exactly this case was not quite
what caused me to think of the question, it was one of the puzzling cases
that I thought of in connection with it.

Let me now try to explain why I don't think {*} above captures my concern.
I claim (uncontroversially, I assume)

[*] If neg ConT were provable in T, then T would be inconsistent.

The argument for this is simple: 'neg ConT <--> neg G' is provable in T; if
neg G were provable in T, then T would be inconsistent; hence, if neg ConT
were provable in T, T would be inconsistent.

Query: Does {*} above adequately capture [*]?

I think not ... at least not if what one means by that is that asking after
the truth of [*] is rightly represented as asking after the provability in
T of 'Prov_T(neg ConT)-->neg ConT'. It couldn't be ... because [*] is
clearly true and provable (in informal thought) while 'Prov_T(neg
ConT)-->neg ConT' is not provable in T. So, asking after the truth of [*]
is not the same as asking whether 'Prov_T(neg ConT)-->neg ConT' is provable
in T.

So, my question is not whether, for s unprovable in PA, 'Prov-PA(s)-->s' is
provable in PA. That has a clear answer ... given above.

It is rather just what I said: for s unprovable in PA, is it always the
case that PA would be inconsistent if s were provable in PA. If you say
that is not precise or meaningless, then you must also say that it is
meaningless to say that if G (or not G) were provable in PA, then PA would
be inconsistent. I don't think that this is meaningless ... and I don't see
any reason to do so. In some sense of the term 'precision', it may lack the
precision that the question 'Is 'Prov_T(neg ConT)-->neg ConT' provable in
T?' has ... but that doesn't mean that the one should be substituted for
the other.



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Michael Detlefsen
Department of Philosophy
University of Notre Dame
Notre Dame, Indiana  46556
U.S.A.
e-mail:  Detlefsen.1 at nd.edu
FAX:  219-631-8609
Office phone: 219-631-7534
Home phone: 219-232-7273
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