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Question on Zorn's Lemma

  1. Aug 31, 2011 #1
    I was reading the proof that every Vector Space has a basis which invoked Zorn's Lemma. The proof can be found http://mathprelims.wordpress.com/2009/06/10/every-vector-space-has-a-basis/" [Broken].

    Now I have an issue specifically with the claim that [tex]U := \bigcup_{S\in C}S[/tex] is an upper bound for C. Applying the same idea as the proof, this seems to imply that the natural numbers has a maximal element. Let C be a chain of natural numbers and similarly let us define [tex]A:=\sum_{n\in C}n[/tex] Then [itex]A\in \mathbb{N}[/itex] and is an upper bound for C. Applying Zorn's lemma then implies that the naturals have a maximal element.

    What exactly am I missing here?
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Aug 31, 2011 #2

    Hurkyl

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    No it's not.... (if C is infinite, anyways)
     
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