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Question about the natural numbers.

  1. Jul 8, 2012 #1
    Is the set of natural numbers the only infinite set that is not a power set of another set?
  2. jcsd
  3. Jul 8, 2012 #2


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    Science Advisor

    Hey cragar.

    Well you need to have a building block for numbers in the simplest manner and the natural numbers are that block.

    Do you have any thoughts about a building block that is a subset or rather something simpler than the natural numbers?

    It is a very good question to ask, because these kinds of things get people thinking and understanding and that's always good.
  4. Jul 8, 2012 #3
    why cant we just start with 0 and 1 . and just add 1 to 1 as many times as we want
  5. Jul 8, 2012 #4


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    Staff Emeritus
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    Gold Member

    Nearly every set is not the power set of another set.

    Did you instead mean
    the only infinite sets for which there is not a bijection from X to a power set of another set are sets for which there is a bijection to the natural numbers​
    or maybe
    the cardinality of the natural numbers is the only infinite cardinal number that is not the cardinality of a power set​

    If so, then your question is essentially the Generalized Continuum Hypothesis.
  6. Jul 8, 2012 #5


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    Gold Member

    You might just as well start with 0 and add 1,000,000 and keep adding 1,000,000. You'll end up with a set that is the same size as if you had added 1 each time
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