Cambridge Tripos 2015: Zorn's Lemma Proof Requiring Axiom of Choice

Question

I was doing a Cambridge Tripos, and the task was to prove Zorn's Lemma and show where the Axiom of Choice was assumed and how it fit into the proof. Reading a paper, I found this:

To prove Zorn’s lemma, it will be convenient to assume that we have a a [sic] “successor operation” on $X$ [a partially ordered set], denoted $x\mapsto x^+$, such that $x^+ > x$ if $x$ is not maximal, and $x^+ = x$ if $x$ is maximal. (The axiom of choice guarantees that there is indeed such a function.)

My question is how the Axiom of Choice guarantees the existence of this function.

Answer 1

I am guessing that $X$ is a partially ordered set in this context, and you want $x^+$ to be somehow a larger element in that specific ordering.

Then for every $x$, consider $C_x=\{y\in X\mid x<y\}$. Then $C_x\neq\varnothing$ if and only if $x$ is not a maximal element. Using the axiom of choice, we can therefore have a choice function $F(C_x)\in C_x$. And defining $x^+=F(C_x)$, or $x$ if $C_x$ is empty, is just fine.

Answer 2

I presume $X$ is a partially ordered set. For each nonmaximal $x\in X$, then $A_x=\{y\in X:y> x\}$ is nonempty. But if $x$ is maximal, define $A_x=\{x\}$. By AC then there is a choice function $f:X\to X$ with $f(x)\in A_x$. You could write $x^+$ for $f(x)$.