[FOM] Independence without forcing
E. Todd Eisworth
eisworth at math.uni.edu
Sun Jul 13 21:25:41 EDT 2003
[Reply to Friedman and others]
I think John Steel hit the nail on head as far as what I was originally
trying to ask. In my last post, I wrote down a hypothetical position and
asked if it was refutable. Harvey's work does the job of refuting it
beautifully.
However, consider now a revised position:
(*)
"All statements of 'real mathematics' are decided by the axiom system ZFC +
V=L + whatever large cardinals make sense in L; everything else is
metamathematics."
Is this refutable?
(Lots and lots of things are decided by that axiom system, though maybe they
are decided in an implausible way.)
Where is this leading? I am consistently amazed and thrilled at the ways
which we (the broad community of mathematical logic) have shown that the
answer to the question "Does mathematics need more axioms?" is a resounding
"YES". (I may have never heard of Boolean relation theory before this last
week, but it has been a real pleasure to read Harvey's papers over the last
few days.)
What about the question "Will mathematics always need more axioms?" Will
there come a time when mathematicians agree on the truth of enough axioms so
that we can really say "all non-metamathematical statements are decided by
our axioms", or is there some super-incompleteness phenomenon that says "no
matter how we choose to formalize our axioms and the definition of
'non-metamathematical', there will always be a non-metamathematical
statement not decided by our axioms"?
The position stated above is an attempt to figure out where exactly our
current knowledge of producing independence results peters out.
If we have refutations of (*), does anyone have other suggestions for other
candidates?
Best,
Todd
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