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Prove Zorn's Lemma is equivalent to the following statement

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  1. May 20, 2017 #1
    1. The problem statement, all variables and given/known data
    From Introduction to Set Theory Chapter 8.1 exercise 1.4

    Prove that Zorn's Lemma is equivalent to the following statement:
    For all ##(A,\leq)##, the set of all chains of ##(A,\leq)## has an ##\subseteq##-maximal element.



    2. Relevant equations
    N/A

    3. The attempt at a solution
    (##\Rightarrow##): Suppose Zorn's Lemma holds and let ##(A,\leq)## be a partially ordered set and let C be its set of chains. It's clear that each element of C is bounded above (##X \subseteq A## for each ##X \in C##) and thus has C has a ##\subseteq##-maximal element by Zorn's Lemma.

    (##\Leftarrow##): Suppose, for all ##(A,\leq)##, the set of all chains of ##(A,\leq)## has an ##\subseteq##-maximal element. Let (##P, \leq##) be a partially ordered set, C the set of chains of P, X a maximal element of C, and suppose that every chain of P is bounded above; we show that P has a ##\leq##-maximal element. Since X is bounded above there exists a ##c \in P## such that ##x \leq c## for all ##x \in X##. Now let ##y \in \bigcup C = P## (as the singleton subsets of P are trivially chains of P) such that ##y \not\in X## (if no such y exists then X = P and c is the greatest element of P and hence a ##\leq##-maximal element). Then, if ##c \leq y##, we have ##x \leq y## for all ##x \in X## and thus ##X = X \cup \{y\}## as ##X \cup \{y\}## is a chain of P and X is a ##\subseteq##-maximal element of C, contradicting our choice of y. Hence ##y \leq c## for all ##y \in A## so c is the greatest element of A and thus a ##\leq##-maximal element.
     
  2. jcsd
  3. May 20, 2017 #2
    Assume that
    Then in conditions of the Zorn lemma this maximal chain has an upper bound. This upper bound is a maximal element of ##A##.
     
  4. May 20, 2017 #3
    I see now. If ##y \in A## and ##c \leq y##, then X is a subset of the chain ##X \cup \{y\}## and so ##X = X \cup \{y\}## and ##y \in X## and we must have y = c.
     
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