FOM: sterility?

[Randall Holmes]

Dear Dr. Kanovei,

I reply to your letter point by point.

You said,

a) 2nd order logic axiomatizes N caregorically 
through appeal to subsets of N. 

I reply:

This is not the best way to see it.  I prefer to say that one appeals
to properties rather than sets (because of the "topic-neutral" character
required of a logic).

You said:

b) 1st order logic does it with equal  
elegancy: simply require, in addition to 
the 1st order Peano axioms, that every  
proper nonempty initial segment has a 
maximal element.

I reply:

But your additional condition cannot be expressed in first-order logic --
it is of course easily expressed in second order logic.

You asked: 

Is there any clear, well defined reason why 
2nd logicians consider a) as so remarkably 
superior to b) as to make it the sole 
cornerstone of a competitive system of f.o.m. ? 
Apparently, the only mathematical point here is that 
some properties of mathematical structures S, 
not expressible in the *corresponding* 1st-order 
language, 
become expressible if quantification 
over P(S) is allowed.

I reply:

Precisely.  That is the point.  It's nothing more than that -- and
also nothing less than that.  The aim is to be able to define what is
meant by (for example) the natural numbers (up to isomorphism).  In
first-order logic, this cannot be done.  Since the sequence of natural
numbers is one of the fundamental ideas of mathematics (predating any
attempt at f.o.m.) a foundation for mathematics that doesn't allow the
unique specification of this structure is suspect.

I'd prefer (once again) to talk about quantification over the
properties of elements of the structure S; talk of P(S) is equivalent
in the presence of suitable set-theoretical assumptions, of course.

Final remark:

I'm not sure that I'm talking about a competing system of foundations.
I'm not proposing or supporting an alternative to ZFC and its
relatives as the de facto foundation of mathematics in practice (in
this thread; I am fond of (first-order!) extensions of NFU as working
foundations).  One possible position (I think that this is John
Mayberry's position (?))  which would have very little effect on the
way mathematics looks in practice is to adopt second-order ZFC (and
extensions as desired) as one's working foundation, using the proof
machinery of the first-order theory (which is of course sound for the
second-order theory).  I'm not entirely comfortable with this because
the ontological commitments of second-order ZF are very strong.

I'm just as impressed as anyone else with first-order ZFC as a useful
machine for encoding mathematical ideas and proving theorems.  It is
also fascinating to study the model theory of ZFC.  But it doesn't do
everything that we need for a foundation of mathematics.  Second-order
ZFC does do everything (or at least quite a lot more -- a complete
foundation cannot be expected) -- if one really believes in _all_ the
levels of the Platonic heaven!  I conjecture that one reason that
people feel that first-order ZFC does everything is that they
equivocate between the first-order and second-order theories.

And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the       | Boise State U. (disavows all) 
slow-witted and the deliberately obtuse might | holmes at math.idbsu.edu
not glimpse the wonders therein. | http://math.idbsu.edu/~holmes