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I Set Theory: Separation Axiom and Garling's Theorem 1.2.2

  1. Oct 31, 2016 #1
    I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis" ... ...

    At present I am focused on Chapter 1: The Axioms of Set Theory and need some help with Theorem 1.2.2 and its relationship to the Separation Axiom ... ...

    The Separation Axiom and Theorem 1.2.2 read as follows:


    ?temp_hash=53fb6f84125d61bbacbbce7a9ff5288b.png


    Garling argues that the Separation Axiom needs to be in place before we can prove Theorem 1.2.2 ... ... but I cannot see where the Separation Axiom is needed in the proof of Theorem 1.2.2 ...

    Can someone give a clear explanation of exactly why we need the Separation Axiom in order to prove Theorem 1.2.2.

    Help will be much appreciated ... ...

    Peter
     

    Attached Files:

  2. jcsd
  3. Oct 31, 2016 #2

    fresh_42

    Staff: Mentor

    He uses the separation axiom for the existence of ##b##.
    Can you define ##A## and ##Q(x)## for the usage in the proof of Theorem 1.2.2? This is needed for otherwise ##b## simply couldn't exist, and then the contradiction became meaningless.
     
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