# [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

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

```