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Size of a set.

  1. Aug 15, 2012 #1
    I have attached part of 2 pages from Justin T. Moores dissertation.
    I am wondering why he says every set of size [itex] \aleph_1 [/itex] has measure zero.
    He is probably using some axioms that i am not familiar with. And Im not sure
    what the [itex] k_2 [/itex] is.
    He says this towards the bottom of the page.
    any help will be much appreciated.

    Attached Files:

  2. jcsd
  3. Aug 15, 2012 #2


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    It definitely uses extra axioms, indeed it can't follow from ZFC alone since it would contradict CH, which is consistent with ZFC. Whenever you see something like:

    Theorem: <Statement>

    It means the <Statement> is a theorem of ZFC. Whenever you see:

    Theorem: (<Axiom(s)>) <Statement>

    It means the <Statement> is a theorem of ZFC + the additional <Axiom(s)>. So in this case, the theorem about [itex]\aleph_1[/itex]-sized sets being null assumes [itex]\mathcal{K}_2[/itex]. A quick Google search yields some relevant results. In particular, look at definitions 4.1 and 2.1 here:

  4. Aug 15, 2012 #3
    ok thanks for your help
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