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Proof about size of a union of sets.

  1. Aug 16, 2012 #1
    Lets say I have [itex] \aleph_1 [/itex] numbers of sets that each have [itex] \aleph_1 [/itex]
    number of elements and I want to show that the union of all of these sets has
    [itex] \aleph_1 [/itex] number of elements.
    I start with the line segment [0,1] and shift this line segment up by all the reals from 0 to 1.
    So now we have the unit square. Now we want to show that this unit square can be mapped to [0,1]. So can we use trick where you take the decimal form of a point and expand it to 2 dimensions. [itex] (.x_1x_2x_3x_4......)\rightarrow (.x_1x_3...),(x_2x_4....) [/itex]
    or another thought I had was to take the cantor set and move it around with a set of reals and map each set to a cantor set that was shifted across the real line.
  2. jcsd
  3. Aug 16, 2012 #2
    You seem under the impression that [itex]\aleph_1[/itex] is the cardinality of the real numbers. This is not the case. The cardinality of the real numbers is [itex]2^{\aleph_0}[/itex].
  4. Aug 16, 2012 #3
    I thought [itex] 2^{\aleph_0}=\aleph_1 [/itex]
    if i replace those 2 quantities, is my idea on the right track.
  5. Aug 16, 2012 #4
    Nope. The statement [itex]2^{\aleph_0}=\aleph_1[/itex] is known as the continuum hypothesis. It can be proven nor disproven. So under the usual ZFC axioms, we can't say that these two cardinals are equal.
  6. Aug 16, 2012 #5
    But yes, your basic idea of doing

    [tex]0.x_1x_2x_3x_4x_5x_6...\rightarrow (0.x_1x_3x_5...,0.x_2x_4x_6...)[/tex]

    is a good one.
  7. Aug 16, 2012 #6
    thanks for the correction about the continuum
  8. Aug 16, 2012 #7


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    This is, I think, an interesting thing to think about re CH:

    What happens when you assume ~CH : then you will find sets in your theory with

    cardinality intermediate between Aleph_0 and Aleph_1. You may cut down on the

    number of functions, say . Of course, I'm being a bit loose here.
  9. Aug 16, 2012 #8
    You mean: between [itex]\aleph_0[/itex] and [itex]2^{\aleph_0}[/itex]?? There is nothing between [itex]\aleph_0[/itex] and [itex]\aleph_1[/itex], even without CH.
  10. Aug 16, 2012 #9


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    Yes, between [itex]\aleph_0[/itex] and [itex]2^{\aleph_0}[/itex].
  11. Aug 16, 2012 #10


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    And I hope this does not distract too much from the OP , but, if you think about it,

    in a sense , there are infinitely-many mathematics: for every undecidable statement,

    mathematics branches out into one system in which the undecidable statement holds,

    and another system in which the statement does not hold. Something similar

    happens with models of different cardinalities, say, the reals: which model is the

    real model? We may choose the standard, the non-standard, or models of any

    infinite cardinality per the compactness theorem.
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