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Cardinality of the set of ordinal numbers

  1. Aug 25, 2011 #1
    Does anyone happen to know what the cardinality of the set of ordinal number (transfinite and otherwise) is? A simplified proof would also be much appreciated. Recently I have been very interested in transfinite numbers and the logically gorgeous proofs involved :D
     
    Last edited: Aug 25, 2011
  2. jcsd
  3. Aug 25, 2011 #2

    micromass

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    The ordinal numbers do not form a set. There are too much ordinal numbers in order for it to be a set. Instead, the ordinal numbers form a "proper class".

    Thus, since the ordinal numbers do not form a set, they do not have a cardinality. Likewise, the cardinal numbers do not form a set and thus don't have a well-defined cardinality.
     
  4. Aug 25, 2011 #3
    Can't you make a set out of anything? I already know the ordinals are uncountably infinate, so I just want it one step farther, "how" uncountably infinate.

    PS. I mean the set of the numbers themselves, not the sets they describe.
     
  5. Aug 25, 2011 #4

    micromass

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    No, you can't make a set out of anything! That's the biggest difference between "naive set theory" and "axiomatic set theory". In axiomatic set theory, there are strict limitations on what can be a set and what can't be a set. It turns out that the ordinals do not form a set.

    I don't know how much you know about ordinals, but here's an easy argument why the ordinals cannot be a set:

    Assume that there exists a set A consisting of all the ordinals. Then [itex]\bigcup{A}[/itex] is an ordinal [itex]\alpha[/itex]. But then [itex]\alpha+1[/itex] is an ordinal which is not contained in A.

    The above proof is known as the Burali-Forti paradox. It was one of the reasons that axiomatic set theory was developed.
     
  6. Aug 27, 2011 #5
    Oh. Fun stuff.
     
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