[FOM] Uncountable structures and `core mathematics'

[Mark Steiner]

It seems that foundational studies researchers have been trying to
demonstrate in various ways that foundational studies (including set theory)
are relevant to "core mathematics".  This is either because (a) foundational
results are in some way EQUIVALENT to results that "core mathematicians" get
in other ways, or because (b) foundational results (new  axioms) can
actually yield natural mathematical results which are not provable using
present "core"methods, or perhaps not FEASIBLY provable using present "core"
methods.

With trepidation I suggest to the experts on thiis list another contribution
of foundational methods to core mathematics (there has been such a volume of
posting on this subject that it is entirely possible that this has appeared
already): using foundational methods, one can give EXPLANATIONS of results
which are provable in core mathematics.

This idea occurred to me years ago when I was pondering the notion of what
makes a proof in mathematics explanatory.  I started looking through a
standard book on model theory, and noticed some elementary examples of this.
We have theorems for various algebraic structures, e.g. that a tower of Hs
is an H, yet the generalization of these theorems you won't find in algebra,
because it turns out that the theorem depends on the syntax of the axioms
defining the various algebraic structures: groups, fields, etc., for example
the quantifier complexity of these axioms.  Our imaginary "core
mathematician", who has never heard of the quantifier hierarchy, would not
even be able to state the relevant theorem, though he can prove each
instance of it.  In any case, it seemed to me that it would be correct to
say that the EXPLANATION of the phenomena in question lies precisely in the
quantifier structure: since by manipulating the structure you can change the
theorem.

I hope even if this is not stately perfectly accurately, it still points to
something useful.