[FOM] Does 2^{\aleph_0} = 2^{\aleph_1}?
Andreas Blass
ablass at umich.edu
Wed Oct 24 12:55:43 EDT 2007
John Baldwin asked whether set-theorists interested in cardinal
characteristics of the continuum believe that 2^{\aleph_0} = 2^
{\aleph_1}, whether we have definite views about the `real' size of
the continuum, and whether we envision the study of cardinal
characteristics of the continuum as clarifying the `truth' of the
continuum hypothesis. Here are my opinions on these matters; I don't
claim that others working on cardinal characteristics will agree.
I have no strong opinion about the `real' size of the continuum. I
think it likely that it is either aleph_1 or a weakly inaccessible
cardinal. I'm quite confident that it's not aleph_{73}, even though
that value is consistent with ZFC. I don't see much chance that the
theory of cardinal characteristics will clarify the real size of the
continuum --- a more likely source for such clarification would be
Woodin's Omega-logic program. (This program threatens to make the
continuum aleph_2, not one of my preferred answers.)
Many of the known independence results about cardinal characteristics
depend on Shelah's proper forcing technology, which typically begins
with a model of the generalized continuum hypothesis and then
iterates, for omega_2 steps with countable supports, continuum-sized
proper forcings. This produces models where 2^{\aleph_0} = 2^
{\aleph_1} = aleph_2. The method doesn't work with longer
iterations; after omega_2 steps, cardinals start to collapse. So we
spend a lot of time constructing and studying models where this
equation holds, but that doesn't make me believe that this equation
is "really" true. (Many years ago, Haim Judah, who then worked on
this theory, suggested that we should accept 2^{\aleph_0} = aleph_2
as an axiom, because we have far better technology for constructing
models of it than for alternatives. I didn't buy it; even though I
agree about the technology, I don't see it as a reason to accept
axioms.)
Andreas Blass
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