[FOM] RE: FOM Continuum Hypothesis
Matt Insall
montez at fidnet.com
Mon May 19 13:30:56 EDT 2003
Insall:
> Thanks to Bill Tait for ...
> He also points out that ...
> inside any model of set theory - including those which fail to satisfy
> CH -
> one may find a constructible universe, and it will have the same
> cardinals
> as
> the original model.
Tait:
I hope I'm not being too defensive, but I just want to note that what I
said was that every cardinal in the original model is a cardinal in the
inner model of constructible sets.
Insall:
Not at all. I should have been more careful in representing to the list
your statements to me offlist.
Now, in regard to the principle I mentioned previously (amended):
(*) For n>1, there is a cardinal k>n such that the power set of n has
more than k members.
Principle (*) is actually weaker than I want to assume for my set theory.
Before I state a more radical departure from CH, let me address another
point Tait brought up offlist. He asks why I single out (*), when for
finite cardinals, also the following property holds:
(^) For n>2, there is a cardinal k>n such that 2n>k.
He makes a good point. My answer offlist was that I have already accepted
the axiom of choice, which generalizes a property that is obvious for
finitely
many finite sets to the case of arbitrarily many sets of arbitrary
cardinality,
and the axiom of choice implies that (^) is false for infinite cardinals.
(Actually,
as you can learn from the wonderful compendium of results by Howard and
Rubin,
the following denial of (^) is a very weak form of the axiom of choice: For
each
infinite cardinal n, 2n=n.) I am not alone in having chosen to accept the
axiom
of choice, as is evidenced by the fact that much of mainstream mathematics
now
uses ZFC - as I said before - as its ``proving ground''.
Let me now recall a principle I have previously suggested that is based upon
what happens
in the finite case, even in the presence of the axiom of choice:
(%%) |P(x)| is as much larger than |x| as is theoretically possible.
In this scenario, there are, for any infinite cardinal k, 2^k cardinals
between k
and 2^k. This very strongly rejects GCH.
References
P. Howard and J. Rubin, Consequences of the Axiom of Choice, Amer. Math.
Soc., 1998.
M. Insall, http://www.cs.nyu.edu/pipermail/fom/2000-February/003814.html
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