FOM: Question grown from Friedman talks JoeShipman at
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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