Re: aleph-null versus aleph-one

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding John's definition:
> ...
>   2 ^ Aleph_null = Aleph_one (definition of Aleph_one).

AFAIK this is *not* the definition of Aleph_one.  In fact, the truth of
the above statement *presumes* the very continuum hypothesis that you wish
to make so explicit.

It is readily proved that 2^(Aleph_null) = c = the cardinality of the
reals (crudely, we can simply consider the set of all the infinite binary
sequences as binary expansions of the real numbers between 0 and 1 by
putting a binary radix point in front of each sequence).

BTW, a nice summary of this stuff is at: <http://www.ii.com/math/ch/> .

>   The real line is sometimes called "the continuum".
>   How many reals are there?  C.
>   C is infinitely bigger than Aleph_null.
>   C - Aleph_null = C.
>   C / Aleph_null = C.

So far so good.

>   C <= Aleph_one.

This seems to be backwards.  It should read: Aleph_one <= c.  The
definition of Aleph_one is that it is the smallest transfinite number
that is strictly greater than Aleph_null (i.e. it is the *next* level of
infinity greater than the number of integers).  The cardinality of the
reals, c is known to be the exponential of Aleph_null.  What is not
proved is whether or not there is a transfinite number that is smaller
than c but greater than Aleph_null.  Presuming that there is no such
transfinite number so that Aleph_one is, in fact, c *is* the continuum
hypothesis.

>   You can assume C = Aleph_one if you like; many people do.
>   This is called the continuum hypothesis.

True.

>   Nobody will ever disprove the continuum hypothesis.
>   Nobody will ever prove it, either.
>   It is provably not provable, just as in Euclidean geometry
>     the parallel postulate is provably not provable.

This undecidability was proved by Cohen in the 60s.  But this has not
stopped *some* people from trying to prove it or disprove it anyway.
Trying to prove or disprove the continuum hypothesis in mathematics (in
spite of Cohen's Field's Medal winning work) is sort of like trying to
build a perpetual motion machine in physics (in spite of thermodynamics).

>   If you invoke the continuum hypothesis, it is good
>     etiquette to say so explicitly.

I suppose.  But it is probably best to properly invoke it when you do
decide to do so.

Actually, I suspect that Larry's question about how much larger the
number of reals is than the number of integers (rationals) did not
require the continuum hypothesis to answer.  All that seems to be
required is the exponential/logarithmic relationship between the
cardinality of the continuum and the cardinality of the integers.
Whether or not there are any distinct transfinite numbers strictly
between these two cardinalities is an interesting question, but I don't
think it was asked by Larry.

David Bowman
David_Bowman@georgetowncollege.edu