numbers [was Re: understanding probability]

[D. Hugh Redelmeier]

| From: Jamon Camisso <jamon.camisso-H217xnMUJC0sA/PxXw9srA at public.gmane.org>

| Yes: see Transfinite Cardinal Arithmetic with Wolfram|Alpha
| 
| http://blog.wolframalpha.com/2010/09/10/transfinite-cardinal-arithmetic-with-wolframalpha/

Nice, easy article.

I always thought that aleph-sub-0 was pronounced "aleph-null", not
"aleph-zero".  Maybe that's old school.

" Kurt Gödel showed that the known axioms of mathematics are
insufficient to disprove the hypothesis. This seemed tantalizingly
close to a proof of the continuum hypothesis, but then in 1963, Paul
Cohen showed that the known axioms are also insufficient to prove the
continuum hypothesis. In other words, the known facts of mathematics
are simply not sufficient to answer Cantor’s question."

The summary seems wrong-headed.  Axioms are not well described as "the
known facts".  They are the postulates, the only givens, the starting
point.  So we cannot discover anything to prove or disprove the
continuum hypothesis.

Finding a distinct number between aleph-0 and beth-1 would constitute
a disproof.  So it is clear that we cannot find such a number.  So
looking is pointless.

There are three paths forward, and they can all be pursued (I think
they are):

(a) work with the (original) system in which the continuum hypothesis is
    unresolveable.  Most useful work doesn't care.

(b) create a new system with an added axiom equivalent to the continuum
    hypothesis and work with that.  I think that that is fairly common.

(c) create a new system with an added axiom equivalent to the opposite
    of the continuum hypothesis and work with that.  I don't think that
    this is common.

Mathematicians generally work in (a) since almost any result they
prove in (a) is true in (b) and (c) as well.  Of course you can prove
in (a) and not in (b) or (c) that you cannot prove or disprove the
continuum hypothesis, so not every result carries over.

When mathematicians need the continuum hypothesis for some proof, they
work in (b).  I don't think (c) has a rich a set of results for
harvesting.