[FOM] Re:Independence without forcing

E. Todd Eisworth eisworth at math.uni.edu
Thu Jul 10 16:28:50 EDT 2003

Todd Eisworth has asked about independence results in set theory that do 
not use forcing. Harvey Friedman has found numerous examples, mostly of a
combinatorial nature, of propositions that are provable from the existence
of large (but not very large) cardinals together with ZFC.

I've kept up with a tiny amount of Harvey Friedman's work in this area
(mostly things about subtle cardinals and linear orderings). I guess the
easiest answer to my question would be that ZFC + V=L neither proves nor
refutes the existence of an inaccessible cardinal assuming that inaccessible
cardinals are consistent with ZFC.

I am more interested in the hard version of the question --- assuming only
CON(ZFC), how might one go about finding a "mathematical" statement P for
which both CON(ZFC + V=L + P) and CON(ZFC + V=L + not P) hold? Does anyone
have any speculations on a plausible scenario for establishing such results?
Any guesses on what such a P should look like? Is it a waste of time to be
concerned with such matters?

I suspect that what I am asking for is (given the current state of our
knowledge) the mathematical equivalent of science fiction, but I like to
think science fiction has a hand in the development of science.
My interest in this stems from conversations with Shelah while I was in
Jerusalem.  Some of his speculations are recorded in Section 9 of "On what I
do not understand..." in Fund. Math. (166) 1 1-82. I thought it might be
interesting if others who have thought out such matters shared their



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