FOM: sterility?

[Harvey Friedman]

I am writing a posting on second order logic to further clarify the
situation, and I will post it shortly. In the meantime, I want to reply to
Holmes, 10:03AM 3/23/99 and 10:14AM 3/23/99.

>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.

Holmes uses an inappropriate notion of "expressed" here and throughout his
postings. The appropriate way to look at this is as follows. There is the
first order language of set theory. It is then trivial to express such
concepts as "inductive number system" in this first order language of set
theory. Since there are no axioms and rules of inference involved here,
there is no issue as to "nonstandard models" and the like. And this is the
normal concept of "express" that mathematicians use.

>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.

As I said earlier, Holmes continues to use an inappropriate notion of
"express" which is contrary to the usual notion of express that
mathematicians use.

In fact, you cannot express "inductive number system" in some treatments of
even second order logic since in some treatments the universe may have
exactly one object. I.e., it is not second order valid that there exists an
inductive number system. So what Holmes is saying is totally unclear on
multiple levels.
>
>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

This is very good.

>(in
>this thread; I am fond of (first-order!) extensions of NFU as working
>foundations).

This is very bad.

>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.

If I understand this correctly, it would be trivially equivalent to
adopting first-order ZFC (and extensions as desired) as one's working
foundation. I don't see the difference. If you do, spell out the
difference.
>
>I'm just as impressed as anyone else with first-order ZFC as a useful
>machine for encoding mathematical ideas and proving theorems.

Very good.

>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.

What is missing? Of course, large cardinals and other related issues are
missing. But what are you referring to?

>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.

What on earth does second-order ZFC do that first-order ZFC does not do for
f.o.m.? Perhaps you should explain what you mean by second-order ZFC.

>You said:
>
>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:
>
>No, not expressible in _any_ first-order language.

As I emphasized above, the first order language of set theory is extremely
- perhaps overly - expressive.

>The property of
>being a model of true arithmetic is not expressible at all in any
>first-order theory, however strong.

It is trivially expressible in the first order language of ZFC.

>(In the following sense: any
>first-order theory which describes a certain structure S in its domain
>in terms compatible with S being a model of true arithmetic has models
>in which S is in fact not a model of true arithmetic).

This is a wholly inappropriate and irrelevant sense of "express."
>
>The issue here is entirely one of what one can define or express;

This is a nonissue since you can express what you want in the first order
language of set theory.

>one gains nothing in terms of proof machinery which cannot be gained
>by working in a stronger first-order theory.

I don't understand this sentence.

>The point which is
>being belabored is that the reference of mathematical language is
>an important foundational issue.

OK, let's talk about it more sensibly.