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Hartog's theorem

  1. Jan 6, 2010 #1
    Write out the proof of Hartog's Theorem again carefully highlighting how the Axiom of Replacement is used

    How can you highlight the axiom of replacment?
    Last edited: Jan 7, 2010
  2. jcsd
  3. Jan 6, 2010 #2
    In the proof of Hartog's theorem, you reach a point where you just proved that the class of all well-orders of the set X (using the separation axiom schema). Now you have to prove that each the class of all order-types (isomorphic ordinals) of each of these well-orders is also a set, and it's here that you use replacement, by constructing a first-order formula that expresses the isomorphism of each (Y,<), where Y is a subset of X and "<" is a well-order in Y, to an ordinal.
    This is the key step in Hartog's proof, the one that allows him to sidestep AC, so I think the "highlighting" is a careful, step-by-step explanation of how the replacement schema is used.
  4. Jan 7, 2010 #3
    That makes sense, but how do you construct a first order formula?
  5. Jan 7, 2010 #4
    I assume that, if you're studying formal Set Theory, that you are familiar with first-order logic (otherwise, it's impossible to understand anything of Set Theory).
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