[FOM] AXIOM SCHEMATA

[Neil Tennant]

On Thu, 15 Jul 2004, Donald Stahl wrote:

> Assuming that Tennant's questions both refer to the preference (not the 
> tendency) 

Yes, they do ...

> I suggest (not on my own behalf) that it might have to do with the 
> idea that compositionality is supposed to be the source of novelty and 
> unlimitedness---that these are to be worked for, not assumed.  Perhaps it 
> has to do with conflating theories with languages, so that the set of axioms 
> corresponds to the lexicon. A non-finitely axiomatized theory would then be 
> on a par with an unlearnable language.

I don't see that the two would be on a par at all. On a par with respect
to what?---learnability?

But that can't be right. First-order Peano arithmetic is learnable, but
not finitely axiomatizable. All that is important is that we should have
an effective method for telling, of any sentence in the first-order
language of arithmetic, whether it is an axiom. The only advantage of a
finitely axiomatized theory is that the effective method in question will
be more efficient than the effective method for an infinitely axiomatized
theory. There does not seem to be any *epistemological* advantage for
finitely axiomatized theories over the various naturally axiomatizable
(but not finitely axiomatizable) theories in mathematics. This is because
the latter have at most finitely many axiom schemata; and for each such
schema, the epistemological problem of warranting its instances is uniform
across them. (The second-order theorist would argue that this is because
we can directly grasp the truth of the second-order quantified sentence
for which the axiom-schema at first order is doing weaker duty.)

Neil Tennant