Difference between Zorn's Lemma and the ascending chain condition

Question

Let $S$ be a non-empty partially ordered set with respect to a relation $\leq$. Then:

Zorn's Lemma: If $S$ has the property that any totally ordered subset $U\subset S$ has an upper bound, then $S$ has a maximal element

Ascending Chain Condition: Every strictly increasing chain eventually stabilises.

The a.c.c implies that every subset of $S$ has a maximal element and we can also deduce the same from Zorn's Lemma, so what, if any, is the difference between the two? I feel like they are both saying the same thing

Answer 1

This is much weaker.

Consider the natural numbers, with an added maximal point which we shall call $\infty$. This partial order satisfies Zorn's lemma, every chain has an upper bound, indeed $\infty$ itself is an upper bound (it's a maximum element!) of any chain.

But the chain $\Bbb N$ is strictly increasing and does not stabilize. So this linear order does not satisfy the ascending chain condition.

The two statements are very close, and for a good reason. Stating that "the ascending chain condition implies the existence of maximal elements" is equivalent to stating Zorn's lemma when the chains are finite. Namely, "if every chain is finite then there exists a maximal element". This is also equivalent to the Principle of Dependent Choice which is a weak version of the axiom of choice (although strong enough to prove countable choice).