FOM: a blunt apology and a sharpened question

[Neil Tennant]

My ears are red. Both Harvey Friedman and Bill Tait have pointed out
the obvious Craig-style trick to turn any axiomatization into one
whose axioms are all logically falsifiable. I hereby publicly kick
myself for not seeing the answer... (I apologize also for the
redundancy in the question---"consistent" plus "true in M"---which
arose from careless editing upon re-formulation at one stage.)

So the existential question is answered positively; it remains to see
whether we can sharpen the original question so that this easy
positive answer would be ruled out in a non-ad-hoc way.

But instead of putting forward another ham-handed formalized statement
that will probably end up being proved or refuted while you do the
crossword over breakfast or put the coffee beans in the grinder,
perhaps I should let co-FOMers in on the more philosophical motivation
behind my question.

The Kantian view of mathematics was that it was synthetic a priori. A
first approximation (even if not exegetically correct) of "synthetic"
would be: not logically true (with a suitably broad construal of
"logic" so as to include perhaps various meaning postulates). So a
Kantian would be interested in having *synthetic* axiomatizations for
a branch of mathematics such as arithmetic or geometry. That is, every
axiom should be logically falsifiable. (The Kantian would also be
interested in methods of proof that prevented "lapses into logical
truth" as one worked from logically falsifiable premises to the
conclusion; but that is another story.)

At the same time, such an axiomatization should be mathematically
pleasing in other more usual respects. One should strive for as much
independence among the various axioms as possible. One should try to
express them as succinctly and elegantly as possible. Clearly a list
of axioms whose members were of the form

	P&A1, P&A2, P&A3, ...

would not satisfy these rough and ready criteria if {A1, A2, A3, ...}
had the same logical closure.

But the complaint that the "P&Ai" axiomatization was not independent
enough would not cut much ice if the "Ai" axiomatization were itself
not completely independent. 

Another way, perhaps, to object to the "P&Ai" axiomatization would be
to focus on the status of P within the theory. P was chosen as a
logically falsifiable sentence of the theory. If P (on the original Ai
axiomatization) was not self-evident, and had to be proved with some
trouble from various Ai, with what right do we now give it the
one-step proof by &-elimination from any of the new axioms P&Aj? The
problem now, however, would be to explicate, in logical terms, what
would be meant by "self-evidence"; and this hardly seems tractable,
however intuitively appealing the notion might be.

Yet another strategy might be to object that P really "does no work"
(or need not do any work) within the "P&Ai" axiomatization. Every
appeal to any axiom is (or could be turned into) a matter of
extracting the right-hand conjunct Ai by &-elimination, and ignoring
the left-hand conjunct P. So the internal logical structure of P need
not contribute anything to the ensuing reasoning. Making this
objection stick, however, faces some difficulties, since someone
*could* give a proof within the "P&Ai" axiomatization by extracting P
by &-elimination and really exploiting its internal structure in order
to obtain some other result (mimicking, say, the further development
of the area when P was perhaps a useful lemma drawn from the earlier
Ai axioms).

So I am left with a rag-bag of hard-to-formalize intuitions
(independence or non-redundancy; being needed for proofs; doing real
logical work; succinct and elegant; self-evident axioms; epistemically
valuable proofs...) about properties that I'd *like* to see suitably
sharpened, and in such a way that the original question, if it admits
of a positive answer in the sharpened formulation, will have a
*non-trivial* answer.

The question would take the [not yet fully sharpened] form:

Given any decidable set X_0 of axioms that are [self-evident about M;
non-redundant,...etc.], is there a decidable set X of *logically
falsifiable* axioms that are [self-evident about M ...etc.] such that
X proves everything that X_0 does, and indeed does so by means of
proofs that are epistemically just as valuable as the ones from X_0?

Any suggestions of where to start looking for sharpenings and possible
answers would be most welcome. I'm basically asking for mathematical
logic to come to the aid of philosophical intuition, or to show that
the intuitions themselves simply cannot be satisfactorily explicated,
or are in some sense incoherent.

Neil Tennant