FOM: Where stands CH?
sf at Csli.Stanford.EDU
Mon Nov 24 21:45:35 EST 1997
In his posting of Nov. 19, John Steel raised several objections to my
paper "Does mathematics need new axioms?" that I plan to take up
separately, beginning with the last one. He protested my belief that CH
is an "absolutely undecidable" proposition in some sense. Actually, what
I said is that I believe CH to be an inherently vague problem that no new
axiom will settle in a convincingly definite way. I would not at all say
the former, since that implicitly treats CH as a definite proposition.
(Steel admitted not having read my paper, and relied instead on having
heard two public presentations of it; but I'm surprised he didn't remember
my use of the words "inherently vague" concerning CH, that I have also
used in his presence on other occasions.) I can't claim to justify my
belief by means of a precise definition of "inherently vague", and so it
is reasonable to object that my belief itself is vague. True. All I'm
reporting is a gut feeling, but not necessarily to be dismissed because of
that. I also think a lot of other set-theoretical statements are
inherently vague in the sense that they do not express definite
propositions; CH is just the most prominent of these, and is of course the
first interesting problem in cardinal arithmetic, if one takes the notions
of cardinal arithmetic at face value.
Steel says that "we understand a lot more about the CH than we did 30-35
years ago, and that large cardinal hypotheses have played an important
role in that increased understanding. There are, for example, the forcing
axioms (PFA and Woodin's P_max axiom which decide CH negatively..."
He also mentioned Woodin's theorem that CH is some kind of "complete
invariant" for the Sigma^2_1 theory of the reals, granted the existence of
a measurable Woodin cardinal. In what sense should we accept these?
Simpson also asked earlier what I had to say about P_max re CH. Well, I
didn't hear a big "whoopie" from the experts to tell the world that now CH
had been settled for one of the above reasons, the way the experts told us
that now Fermat's last theorem had been settled. Who thinks we will
recognize a solution of CH when it is presented to us? Just a gut feeling...
PS. Of course, we will have to rely on the experts to proclaim the
solution if it is ever to be proclaimed, at least at first.
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