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Question about cardinality and CH

  1. Oct 31, 2010 #1
    Under the continuum hypothesis, we readily see that that [tex]|{a < \aleph_1 : \textrm{a is a cardinal}}| = \aleph_0 [/tex]. What happens under the negation of CH? Is this equality still true or not? If the latter, always under the negation of CH, are there any infinite cardinals lambda for which the inequality [tex]|{a < \lambda : \textrm{a is a cardinal}}| = \lambda [/tex] fails?
     
    Last edited: Oct 31, 2010
  2. jcsd
  3. Oct 31, 2010 #2
    First, what do you mean with [tex] |a<\aleph_1 : a \text{ is a cardinal}|[/tex]. Do you simply mean the cardinality of the set [tex] \{a<\aleph_1 : a \text{ is a cardinal}\}[/tex]??

    In that case, it is always true that

    [tex] \{a<\aleph_1 : a \text{ is a cardinal}\}=\mathbb{N}\cup \{\aleph_0\} [/tex]

    So wheter CH holds or not, this set is always countable.
    In general we have that (by definition almost)

    [tex] \{a<\aleph_\alpha : a \text{ is a cardinal}\}=\mathbb{N}\cup \{\aleph_\beta~\vert~\beta<\alpha\}[/tex].

    For your second question, this is not always true. For example:

    [tex] \{a<\aleph_2 : a \text{ is a cardinal}\}=\mathbb{N}\cup \{\aleph_0,\aleph_1\}[/tex].

    But this is also countable, so [tex] |a<\aleph_2 : a \text{ is a cardinal}|=\aleph_0[/tex]

    Where CH does come in to play, is with the set

    [tex] |a<2^{\aleph_0} : a \text{ is a cardinal}|[/tex]

    If CH is true, then this is [tex]\aleph_0[/tex]. But if CH is not true, then it can be any cardinal. E.g. it is consistent with ZFC that

    [tex] |a<2^{\aleph_0} : a \text{ is a cardinal}|=\aleph_{666}[/tex]

    and so on if you replace 666 with any ordinal.

    I hope this answer was helpful...
     
  4. Oct 31, 2010 #3

    CRGreathouse

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    I don't think you can replace it with any ordinal. What about, e.g., [tex]\omega_1[/tex]? Could that really be consistent?
     
  5. Oct 31, 2010 #4
    But it is consistent with ZFC that [tex] 2^{\aleph_0}=\aleph_{\omega_1} [/tex]. See Easton's theorem...
     
  6. Oct 31, 2010 #5
    I believe there are some very weak limits on possible cardinalities of the continuum - I think the cofinality of the continuum is uncountable, and so, for instance, 2^aleph_0 can't be aleph_omega.
     
  7. Oct 31, 2010 #6
    Many thanks micromass, you're clarified a few things for me. :smile:
     
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