FOM: December 1 - December 22, 1998
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FOM: mathematical certainty
Andreas Blass' recent thoughtful note, on a matter I've thought quite a lot
about, raises a couple of questions.
(1) The first (a small point?) concerns his claim that
"As soon as I see that ZFC is [an appropriate formal system for
establishing facts with mathematical certainty]; i.e., that I can be
certain about the correctness of a statement once I have seen it proved in
ZFC, I also see that the consistency statement Con(ZFC) is mathematically
certain, even though it is not provable in ZFC."
This doesn't seem quite right. There is, after all, the matter of whether
the formula Con(ZFC) expresses the consistency of ZFC. So, at the very
least, it seems that the statement needs to be amended to read: 'As soon as
I see that I can be certain about the correctness of a statement once I
have seen it proved in ZFC, and I can see (with mathematical certainty ...
whatever that may mean) that Con(ZFC) expresses the consistency of ZFC, I
also see that Con(ZFC) is mathematically certain'. But seeing that Con(ZFC)
expresses the consistency of ZFC may require substantive knowledge beyond
that which is properly codifiable in ZFC. Indeed, it may require adversion
to evidence that is not 'mathematically certain' (again, whatever that may
mean) at all.
(2) Another (more fundamental?) concern that has vexed me for quite some
time is this: What exactly is the reasoning that is used in establishing
the consistency- (and similar) reflection(s)? There seem to be at least two
Possibility 1: T (a set of axioms) is true. T couldn't be true unless it
were consistent. Therefore, T is consistent.
Possibility 2: T. If T, then T is consistent. Therefore, T is consistent.
[N.B. There is the possibility of a third possibility that would have
something to do with constraints on what a philosopher might term
'epistemic rationality'. It goes something like this: I accept T and I
realize that I accept T. It is not rational for me to accept T unless I
also accept that T is consistent. Therefore, I must accept that T is
consistent. I do not take this model to be as basic as the other two,
though, since I cannot see how to establish the second element of this
reasoning (i.e. the claim 'It is not rational for me to accept T unless I
also accept that T is consistent.') without adverting to either the second
element of Possibility 1 or the second element of Possibility 2. Neither,
on the other hand, do I see how to show that such appeal is inevitable.]
Possibility 1 involves the introduction of a truth concept. Possibility 2,
apparently, doesn't. For non-finite cases of T (e.g. the axioms of ZFC), it
seems that use of a truth concept (or something equivalent to it) is
necessary since we cannot assert or affirm infinitely many statements in a
single act of thought. We need some device for collecting infinitely many
distinct statements into a single assertoric unit. The truth concept is
what we use to do this. In finite cases of T, however, there is no simlar
need for a truth concept. We can simply assert the finitely many statements
making up T.
This difference makes the second premises of the reasonings in Possibility
1 and Possibility 2 different. In Possibility 1, it is the claim 'Every t
in T is true==>T is consistent'. In Possbility 2, it is the claim 'T==>T is
consistent'. G2 seems to clash with this latter (assuming expression of 'T
is consistent' by any of a certain class of formulae Con(T), and assuming
that any such formula satisfies certain certain postulates or conditions
(e.g. the Hilbert-Bernays Derivability Conditions) taken to be at least
partially constitutive of its ability to express the consistency of T). G2
does not, however, clash with the former. The second elements of the
reasonings in Possibilities 1 and 2 thus seem to be different. What exactly
is this difference, and is it a deep difference?
(3) A related matter (raised mainly for the finitely axiomatizable cases of
T mentioned above). If, a la G2 and certain (commonly accepted?)
assumptions concerning the logic of T, T doesn't logically imply Con(T),
what exactly is the nature of the entailment between T and Con(T)? Does the
fact that this entailment cannot be codified in the logic of T raise
questions concerning the adequacy of that logic--in particular, does it
raise legitimate questions concerning the completeness of that (or any
replacement) logic? If so, then what of ultimate interest or value do the
standard completeness proofs that can be given for some of these logics
really tell us? And, if not, then what kind of entailment IS involved? Are
all the elements that may go beyond standard logic attributable to what is
required in order to establish that Con(T) expresses the consistency of T?
Or is the entailment between T and 'T is consistent' (the informal
metamathematical statement, as distinct from the T-theoretic formula
Con(T)) itself a less-than-purely-logical entailment?
Department of Philosophy
University of Notre Dame
Notre Dame, Indiana 46556
Office phone: 219-631-7534
Home phone: 219-232-7273
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