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Is this correct? I think it is wrong. According to wikipedia

Suppose a partially ordered set P has the property that every chain (i.e. totally ordered subset) has an upper bound in P. Then the set P contains at least one maximal element.

However, can't see how this is equivalent to the definition given in notes.

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  • $\begingroup$ Your confusion might come from the fact that you are thinking of a defintion of inductive set that is not meant in this context. You are probably thinking of sets that contain 0 and are closed under successors. Here inductive partially ordered set certainly means partial order in which every chain has an upper bound. $\endgroup$ – Stefan Geschke Dec 29 '11 at 11:29
  • $\begingroup$ -1 for not citing the source for the text you speak about. $\endgroup$ – Henning Makholm Dec 29 '11 at 23:07
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To understand the statement of the theorem you would have to look up what an inductive partial order is. If you do, you should find that it is a partial order in which every chain has an upper bound. Therefore the statements are the same. (I do not have access to your notes to know whether the meaning of inductive was made clear there.)

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Yes, that is the statement of Zorn's Lemma. If S is a partially ordered set in which every chain has an upper bound (i.e. if S is an inductive set and is partially ordered) then S has a maximal element. Your notes say exactly what Wikipedia is saying.

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