Re: CH yet again.

MoeBlee wrote: 
Aatu Koskensilta wrote:

The formulations "c = aleph_1" and "every infinite set of reals is
either countable or has the cardinality of the continuum" are not
equivalent without choice. The first of these obviously implies the
second, but for the other direction you need choice. I don't know
whether this is what Bill had in mind.


Thanks, Aatu. I understand how the first implies the second. But can
you give a synopsis of how the axiom of choice is used to prove that
the second implies the first, or is the proof too complicated to
synopsize in a paragraph?

The usual proof goes like this: by choice the cardinality of the continuum is an aleph, and if there is no cardinality between aleph_0 and the cardinality of the continuum, then the cardinality of the continuum is aleph_1 (since it is greater than aleph_0 by Cantor's theorem). Without choice, the cardinality of the continuum need not be an aleph, and this argument breaks down.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

Relevant Pages

  • Re: Mapping ordinals to the reals
    ... rationals and uncountability of the reals. ... just countably many infinities or uncountably many ... of the collection of curves to get a larger cardinality. ... conclusion of continuum many. ...
    (sci.math)
  • Re: Mapping ordinals to the reals
    ... rationals and uncountability of the reals. ... just countably many infinities or uncountably many ... of the collection of curves to get a larger cardinality. ... conclusion of continuum many. ...
    (sci.math)
  • How many continuous functions? (History)
    ... the earliest appearances of the fact that the cardinality ... We can also conclude that the aggregate of all functions ... the totality of all integrable functions of one or several ... Cantor assumes the continuum hypothesis in some of his ...
    (sci.math)
  • Re: Non-aleph cardinals in set theory without axiom of choice?
    ... continuum, could not equal aleph_1, since it's not an aleph! ... formulations of the continuum hypothesis: ... "The cardinality of the continuum is aleph-1." ... Even on the assumption that the beth-1 is not an aleph ...
    (sci.math)
  • Re: Baire Space - omega_omega
    ... It is consistent with the usual axioms of set ... theory that the cardinality of the continuum is pretty much any aleph. ...
    (sci.math)