FOM: the role of informal concepts in evaluating f.o.m. research

Stephen G Simpson simpson at
Sat Sep 5 00:20:43 EDT 1998

Joseph Shoenfield writes:
 > I have suggested that Steve should defend his position by proving
 > mathematical theorems which clearly show the importance of fcp [=
 > finite combinatorial proposition] in fom [ = f.o.m., foundations of
 > mathematics].

This isn't how I have chosen to defend my position.

Rather, I have defended my position by pointing out that a key
research program in f.o.m. is to extend the incompleteness phenomenon
into various fields of mathematics.  Let's call this the INC-MATH
program.  Martin Davis has called it the G"odel's legacy program.  A
fuller statement of the program is: to devise reasonably natural
propositions which resemble familiar theorems or are stated in terms
of familiar concepts from various fields of mathematics, but which can
be shown to be independent of ZFC or other robust axiom systems.

One of the relevant fields of mathematics is finite combinatorics.
Obviously, the corresponding part of the INC-MATH program involves
finite combinatorial propositions.  Thus the notion of "finite
combinatorial proposition" is essential for evaluating progress in
this part of INC-MATH program.

That is how I defend my position.  Which part of this defense don't
you agree with?

-- Steve

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