FOM: Are Harvey's postings "Foundational"?
heck at fas.harvard.edu
Wed Mar 27 15:26:42 EST 2002
Here is what I take to be the foundational significance of Harvey's
recent work. I could, of course, be wrong, but wanted to say a word so
that it would not appear that Harvey was the only one who had any sense
of what he is trying to do. I should say that my understanding of the
matter benefitted from second-hand information gleaned from Warren
Goldfarb, after Harvey visited gave a talk here last year.
Many mathematicians who do non-foundational mathematics take Goedel's
theorem and related results to be of no real mathematical significance.
Their reasons for this view are somewhat hard to fathom, and I'm not at
all sure there is a coherent position that underlies it. But
nonetheless, that is what they think. The questions set-theorists and
the like study, especially regarding large cardinal axioms and the like,
are regarded as fruity. One could try to convince them that there isn't
a coherent position there, but I doubt one wouold have much success.
What Harvey is trying to do is to show that there are /natural
mathematical problems/, that is, problems that arise naturally within
non-foundational mathematics, that are unsolvable on the basis of, say,
ZFC. It is even better if these problems have fairly natural-seeming
proofs whose /formalization/ requires strong assumptions. The proofs
that the problems in question are unsolvable are, of course, going to be
complicated. But the problems are supposed to be simple, and natural.
Moreover, these problems are often generalizations of yet simpler
problems that /are/ solvable within ZFC, or PA, or what have you. That,
too, is meant to be part of the charm.
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