Plausibility argument for Zorn's Lemma

Solution 1:

This is not quite a mathematical argument. And since it is not really a proof, Zorn's lemma doesn't quite follow from it. What it is is an intuitive explanation trying to convince the reader to accept Zorn's lemma as an axiom.

The point that the paragraph is trying to make is that if we have a partial order satisfying the conditions in Zorn's lemma, then we can define this recursive process which will construct a chain, and if there is no maximal element then this process cannot terminate.

But when we say "terminate" we don't mean that in the "applied" kind of way. We mean that in the set theoretical way. Saying that we in fact defined an injective function from a proper class into a set, which is a contradiction. That what it means "go through infinity, an infinite of infinites etc.", that we go through the first $\omega$ steps, but we can continue, and we go through more and more and more steps and so on.

One reason to give this sort of... half-assed explanation lies in the third word of your question "Physics". To fully understand Zorn's lemma one has to understand some set theory first, to understand the axiom of choice, to understand the well-ordering principle, and so on. One can just write it up as a black box, or convince themselves that they understand how Zorn's lemma works. But I won't expect a physicist to sit down and understand the mechanics of transfinite induction, and the axiom of choice.

So in order to convince a physicist that Zorn's lemma is plausible, one has to resort to furious handwaving like in the quoted paragraph.

The paragraph is, in fact, a broad stroke of how to prove Zorn's lemma from the axiom of choice, using transfinite recursion. The "through infinity" part is what is known as a limit stage in transfinite recursion.

Finally, let me point that of course one has to be convinced about the plausibility of Zorn's lemma. It is unprovable without the axiom of choice (and indeed assuming Zorn's lemma we can prove the axiom of choice) and that makes the lemma very non-constructive. And one can see that from the formulation of the lemma. There exists a maximal element. We are not told how that element looks like, or what sort of properties it may have. It just exists.

Solution 2:

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).