[FOM] Re: Arithmetic-free theory of formal systems?

[Timothy Y. Chow]

On Sun, 23 May 2004, Vladimir Sazonov wrote:
> See "On feasible numbers" in
> http://www.csc.liv.ac.uk/~sazonov/papers.html

I have taken a look at this now and it seems like a promising piece of 
work.  The work mentioned by Jean Paul Van Bendegem sounds related.

> Why do you think that any formal system, even based on
> the first order *language* (and on some weak version of first
> order logic) should have a model *in* "the" universe for ZFC?

Not having read your paper when I posted my question, I did not realize 
that you were using the term "consistent" to mean what a classical 
mathematician might describe as "having no feasible inconsistency."

> By the way, completeness theorem is, *in a sense*, evident:
> if a formalism is consistent then we can imagine corresponding
> "universe" and have some image in our minds. This image serves
> as a model. Of course, some generalization of formally provable
> completeness theorem (not in ZFC) is desirable (if possible at
> all) for formalisms like that for feasible numbers.

I'd agree with this, except that I see no reason to reject ZFC a priori.  
What you'd have to modify is the usual concept of a model.  I would guess
that whatever you come up with could be formalized in ZFC; after all ZFC
is capable of formalizing all kinds of disparate concepts.  It might not
be very *natural* to formalize it in ZFC, but that remains to be seen.

> Inconsistency of PA (if possible at all) would, I believe, lead to
> strong doubts on uniqueness of the meaning of "and so on". So the
> idea of "standard" N (unlike the vague idea on N) will vanish
> (lose a lot of its defenders). Note, that these doubts exist anyway,
> without any contradiction in PA. How can these doubts (which you
> seemingly almost accepted?) be consistent with platonism?

I'm wary of getting drawn into another interminable debate in the 
philosophy of mathematics, but since the insight I recently gained
was valuable to me, I will try to give a brief reply to this question.

Consider the sun.  I believe that the sun is real, that it exists, and
that it is unique.  But what is the mass of the sun to the nearest 
kilogram?  I readily concede that "the mass of the sun to the nearest
kilogram" may not even be a coherent concept, and even if it is, we
may never be able to learn its value.  I also readily admit that our
concept of the sun has evolved through history, and that a modern solar
physicist has a much more sophisticated concept of the sun than our 
Stone Age ancestors did.  None of this causes me to doubt that the sun
is real, that it exists, and that it is uniquely determined.

I do not intend this to be a positive defence of platonism, just an 
observation that accepting the vagueness and fluidity of the concept
of N does not force one to abandon platonism.

Tim