Maximal element of a linearly ordered set
First, this is the definition of Zorn's Lemma in my text. (Just in case it's not a standard way of defining it.)
The problem is:
If A is a non-empty partially ordered set such that every chain in A (a sequence
) has an upper bound in A then A contains a maximal element.
So. The proof.
be a linearly ordered set. The immediate successor of
(if it exists) is the least element in the set
. Prove that if A is well ordered by
, then at most one element of A has no immediate successor.
If is a linear order then by Zorn's Lemma all chains (a, b, ..., z) in A have a maximal element. Since all elements of A are comparable in a linearly ordered set (and therefore all elements of A form a single chain) then there exists a single element z in A such that a < z for all . Thus z has no immediate successor. That is is empty.
I think the proof looks good, but the trouble is I have what I think is a counter-example. Define the set A to be the natural numbers, N, linearly ordered by the usual definition of on the real numbers. I can find no (infinite) chain in N that has a maximal element.
My proof evidently missed something, but I can't tell what is missing.