Re: Can FOL formalize all of math?
- From: herbzet <herbzet@xxxxxxxxx>
- Date: Sat, 20 Jun 2009 23:00:12 -0400
Chris Menzel wrote:
herbzet said:
Chris Menzel wrote:
Dan Christensen said:
Is it correct to say that the theorems of FOL do not cover all of
mathematics?
Well, the answer to that question is *obviously* yes, since the
*theorems* of FOL, being completely uninformative logical truths,
don't (in any reasonable sense) "cover" *any* mathematics.
That was my first thought (well-spoken on your part, btw, as ususal)
but on second thought, the notion of "covering" mathematics is pretty
vague.
Any mathematical proof (rather than theorem) that can be cast in
first-order logic corresponds to a theorem of FOL. In this sense,
the theorems of FOL do "cover" a lot of mathematics.
Sure, but it seems to me that a big part of "covering" mathematics is
identifying the relevant axioms and definitions for the various
branches. The theorems of FOL in the language of PA "cover" classical
arithmetic in your weaker sense, but we have learned nothing of *what*
has been covered until we make the axioms of PA explicit.
Right.
An educated guess is that what DC was actually asking about, is
whether there is generally accepted mathematics that can't be
shoe-horned into first-order form, for example because first-
order languages aren't expressive enough for some mathematical
propositions to be stated in them, or because first-order logic
isn't strong enough to get from some expressible sentence A to
some expressible sentence B.
I think the second possibility can be ruled out on grounds of
completeness: if a set of formula Gamma imply B, there is a
deduction from Gamma to B.
Another possibility for what he's asking about would be for
example the continuum hypothesis. This can certainly be
expressed in first-order language, and it can certainly be
given a formal proof from a first-order set of axioms, if
we're willing to accept some dubious axioms.
I suppose there's the possibility that CH, or other tough
problem, might fall out from some very plausible second-order
axioms that can't be expressed at first-order. But now I'm
just blathering about stuff I don't know anything about.
--
hz
.
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