[FOM] Mathias on the Continuum Hypothesis
E. Todd Eisworth
eisworth at math.uni.edu
Sat Jun 28 17:53:24 EDT 2003
[Mathias ]
It is a remarkable development since Cohen showed CH to be unprovable
that different possible values, among the alephs, for the cardinality
of R no longer seem to be equally interesting or plausible, but that
attention has narrowed to aleph_1, aleph_2 and not-an-aleph as the
three most interesting, even plausible, possibilities.
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I am not sure if I agree with Mathias' assessment. I can't really comment on
the
"not-an-aleph" possibility, but as for the other two, how much of the
"plausibility" is based on the lack of sophisticated techniques for
producing models where the continuum has other values?
The following is taken from Shelah's "On what I do not understand (and have
something to say)" [Fund. Math. 166 (2000) pp. 1-82]
"In fact, the advances in proper forcing make us 'rich in forcing' for
$2^{\aleph_0}=\aleph_2$, making the higher values more mysterious. ... As
we are rich in our knowledge to force for $2^{\aleph_0}=\aleph_2$, naturally
we are quite poor concerning ZFC results. ... We are not poor concerning
forcing for $2^{\aleph_0}=\aleph_1$ (and are rich in ZFC. But for
$2^{\aleph_0}\geq\aleph_3$, we are totally lost: very poor in both
directions."
As a concrete example, why is the statement "the continuum is real-valued
measurable" held to be less plausible or interesting than "the continuum is
of size $\aleph_2$? Is it because we basically know one way of building
models where the continuum is RVM and new techniques seem beyond our reach?
[For those not familiar with the notion, "the continuum is real-valued
measurable" can be taken to mean "Lebesgue measure can be extended to some
measure (not translation invariant) such that all subsets of the unit
interval are measurable". Ulam proved that if such a measure exists, then
$2^{\aleph_0}$ is HUGE. Solovay showed that the existence of such a measure
is consistent relative to the existence of a measurable cardinal.]
I would argue that the preoccupation with the values $\aleph_1$ and
$\aleph_2$ for the continuum exists not because these values are more
plausible or interesting, but rather that these happen to be the values that
arise when we do what we know how to do.
Of course, I'd love to be convinced otherwise...
Best,
Todd
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