Turn off ZeroDivisionError?

Paul Rubin http
Sat Feb 16 06:08:54 CET 2008


Jeff Schwab <jeff at schwabcenter.com> writes:
> > They really do not.  The extended real line can be modelled in set
> > theory, but the "infinity" in it is not a cardinal as we would
> > normally treat them in set theory.
> 
> Georg Cantor disagrees.  Whether Aleph 1 is the cardinality of the set
> of real numbers is provably undecidable.

You misunderstand, the element called "infinity" in the extended real
line has nothing to do with the cardinality of the reals, or of
infinite cardinals as treated in set theory.  It's just an element of
a structure that can be described in elementary terms or can be viewed
as sitting inside of the universe of sets described by set theory.
See:

  http://en.wikipedia.org/wiki/Point_at_infinity

Aleph 1 didn't come up in the discussion earlier either.  FWIW, it's
known (provable from the ZFC axioms) that the cardinality of the reals
is an aleph; ZFC just doesn't determine which particular aleph it is.
The Wikipedia article about CH is also pretty good:

  http://en.wikipedia.org/wiki/Continuum_hypothesis

the guy who proved CH is independent also expressed a belief that it
is actually false.



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