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

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 $C_x=\\{y\in X\mid xeq\varnothing$ 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.