FOM: Question grown from Friedman talks
JoeShipman@aol.com
JoeShipman at aol.com
Mon Feb 5 13:46:58 EST 2001
With regard to independence from ZFC, Friedman sees a big distinction between
open questions which are literally already in the literature, and natural
extensions of such questions. This seems plausible, because any directions
of development for mathematics which would run into independence phenomena
would in my opinion have been abandoned each time they were pursued because
of the difficulty of proving anything. So there's a process of natural
selection going on which is hard to detect by just looking at what's been
published; but I imagine that it must have happened several times that other
mathematicians have formulated for themselves BRT-type generalizations of
parts of ordinary mathematics and not gotten anywhere, and not published
anything. (In other sciences the failure to obtain an expected result can be
publishable, but not in mathematics!)
Of course we are all just guessing here based on vague intuitions; but of the
three most famous open problems, my hunch is that the Riemann Hypothesis is
very unlikely to require axioms beyond ZFC to be settled, the Poincare
conjecture is also very unlikely to, and P?=NP is slightly less unlikely.
My best guess for a famous open problem in "normal" (absolute,
non-set-theoretic) mathematics which requires axioms beyond ZFC is the
Invariant Subspace conjecture.
-- Joe Shipman
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