[FOM] Does 2^{\aleph_0} = 2^{\aleph_1}?
James Hirschorn
James.Hirschorn at univie.ac.at
Thu Oct 25 14:36:40 EDT 2007
On Wednesday 24 October 2007 12:06, joeshipman at aol.com wrote:
>
> >continuum `believed' that 2^{\aleph_0} = 2^{\aleph_1}.
>
> ****
>
> This statement of cardinal arithmetic is considered plausible because
> it is true in the most natural models violating CH -- it follows from
> Martin's axiom, and also from the existence of a real-valued measurable
> cardinal, even though models of these axioms differ strongly in many
> other ways.
Recall that the cardinal characteristic t is the smallest cardinality of a
maximal well ordered subfamily of (P(N)/Fin,\supseteq^*). Thus any "tower"
(a_alpha:alpha < kappa), i.e. a_alpha's are infinite subsets of N,
and alpha < beta implies a_beta \subseteq^* a_alpha, of height kappa < t,
has an infinite b such that b \subseteq^* a_alpha for all alpha.
t is a relevant cardinal characteristic because of the equation
(i.e. theorem of ZFC):
2^{<t} = 2^{aleph-0}.
Martin's axiom implies t > aleph-1 (and hence the equality in question),
whereas RVM implies t = aleph-1. Because of this, and also since adding say
aleph-(omega-1) many random reals to a model of GCH is a basic way of
obtaining 2^{aleph-0} < 2^{aleph-1} + not-CH, I was surprised to hear that
RVM -> 2^{aleph-0} = 2^{aleph-1}. Is there a simple proof of this?
>
> Can anyone point to an example in the literature of a model in which
> 2^{\aleph_0} < 2^{\aleph_1} (or an axiom which implies this) that was
> not explicitly "cooked up" to satisfy this inequality (that is, where
> the inequality follows from other properties used to define the model,
> not directly related to cardinal arithmetic)?
Doesn't V=L qualify? Perhaps you meant an axiom consistent with the negation
of CH?
James Hirschorn
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