[FOM] Nonstandard Methods

[Harvey Friedman]

Reply to Wilson  Nonstandard Models.
On 8/6/03 6:58 PM, "Todd Wilson" <twilson at csufresno.edu> wrote:

Friedman: 
>> I have a radical idea about what to do about such an impasse, and
>> that is to get much more radical than everything else, in a setting
>> where there cannot be any funny business going on.
>> 
>> I.e., do FINITE ANALYSIS ONLY. In fact, do FINITE MATHEMATICS ONLY.
> 

Wilson

> In this connection, I would like to mention an old paper by sometime
> FOM contributor Jan Mycielski:
> 
>   J. Mycielski, "Analysis without actual infinity", JSL 46:3, Sept
>   1981, pp. 625-633.
> 
> Mycielski presents a formal system FIN that is locally finite (every
> finite fragment has finite models) and is "sufficient for the
> development of analysis in the same sense that ZFC ... is sufficient
> for the development of mathematics".  The system includes a
> rationally-indexed set of constants that represent (potentially)
> infinite indiscernables in the theory, the reciprocals of which
> function as infinitesimals, making the development of analysis in FIN
> much like that in nonstandard analysis.

This work concentrates on a formal translation of any first order theory
into a FIN via the Skolem function construction, which is presented as a
sequence of finite stages, each of which is finite.

This is not at all what I have in mind. I was thinking of a major reworking
of standard mathematical concepts in an imaginative but completely direct
fully mathematical way. Detailed estimates would play a central role,
generally involving at most a few exponentials. A vast array of difficult
combinatorial and analytic problems would arise from a systematic treatment,
concerning, e.g., making the estimates reasonably sharp.

No general metamathematical translation could scratch the surface.

For some idea of some relevant metamathematical considerations here, see my
posting 
164:Foundations with (almost) no axioms 4/22/03
http://www.cs.nyu.edu/pipermail/fom/2003-April/006450.html

I conjecture that anything you can do with exotic theories of alternative
mathematics can be done in a way that all mathematicians can relate to in
the explicit detailed concrete finite. I conjecture that, ultimately, the
move to the explicit finite will be the expected norm in mathematics.

What does this do to independence results?

They are nearly entirely preserved. E.g., one should be able to show that
within such detailed finite treatments of mathematics, even within a
universe of finite size, one still has entirely natural statements that can
only be proved using large cardinals, unless one uses at least 2^2^1000000
pieces of paper. 

Harvey Friedman