TITLE: Plausibility argument for Zorn's Lemma
QUESTION [10 upvotes]: In "Mathematical Physics" by Robert Geroch, the following 'plausibility argument' is given for Zorn's Lemma [If every totally ordered subset of a partially ordered set $S$ is bounded above, $S$ has a maximal element]:

It is not difficult to convince onselef that this statement [Zorn's Lemma] seems "true". Pick $s_1$ in $S$. If this $s_1$ is not a maximal element, there is an $s_2$, different from $s_1$, with $s_1 \leq s_2$. If $s_2$ is not maximal, there is an $s_3$, different from $s_2$, with $s_2 \leq  s_3$, etc. Either we eventually obtain a maximal element (in which case we are done), or we obtain $s_1 \leq s_3 \leq s_3 \leq ... $ . In this latter case, $\{s_1,s_2,...\}$ is a totally ordered subset of $S$, whence it is bounded above, say by $s_1'$. If $s_1'$ is not a maximal element, there is $s_2'$ with $s_1' \leq s_2'$, etc. (We "go right through infinity.") If we find no maximal element in $s_1', s_2', ...$, we obtain, again, a totally ordered set, which must be bounded above, say by $s_1''$, etc. Suppose we are unlucky and go through an infinite sequence of such sequences without finding a maximal element. Then we obtain totally ordered $s_{m}^{n} \{m,n = 1,2,...\}$, which must be bounded above, say by $t1$, etc. We "continue in this way. At every stage we have a totally ordered subset, which must be bounded above, so we add in this upper bound to obtain a new totally ordered subset. We can go through infinity, an infinity of infinities, etc. We never get stuck; we can always go on. We should obtain eventually a maximal element."

I don't see how this argument makes Zorn's Lemma plausible, however. If anything, it seems to say that it is possible that we can go through infinities of elements forever, always finding upper bounds and forming totally ordered sets with them, and doing this on and on, without ever finding a maximal element. In particularly, how does the last sentence above "We should obtain eventually a maximal element" at all follow from the argument?

REPLY [4 votes]: The main idea, which is not intuitive at all without some set theory background, is that if this process of going through infinity, infinities of infinities, etc., did not eventually result in a maximal element, then the original $S$ would be too large to be a set (it would be a proper class).