Basically, I'm trying to understand a note in Jacob Lurie's paper On a conjecture of Conway (Illinois Journal of Mathematics 46.2, 2002).
Let $X\subseteq Y$ be sets, $\mathbf{U}$ a proper class, and $\varphi\colon X \to \mathbf{U}$ an injective function. Then in the paper, Zorn's lemma is used on all partial extensions of $\varphi$ to $Y$, i.e. it is used on the collection $$\Phi := \{\psi\colon Z\to \mathbf{U} \;:\; X\subseteq Z \subseteq Y,\; \psi|_X = \varphi\},$$ partially ordered by $\psi \le \psi'$ iff $Z\subseteq Z'$ and $\psi'|_Z = \psi$, for $\psi$, $\psi' \in \Phi$ with respective domains $Z$ and $Z'$.
I have no problem with seeing that every chain in $\Phi$ has an upper bound. The problem is however that $\Phi$ is a proper class (since $\mathbf{U}$ is a proper class), so from my understanding, to apply Zorn's lemma directly, the axiom of global choice is needed.
However, Lurie writes:
The fact that this partial order is actually a proper class introduces a technicality, but it is easy to sidestep since every chain in this partial order is bounded in size.
By that I think he means that global choice is not necessary, but by some "trick" that I do not understand, regular choice is sufficient. Can anyone help me whether this is correct, and if so, point out the trick to me?
(Note that in the paper, $X$, $Y$ and $\mathbf{U}$ are taken to be partially ordered abelian groups, and the considered maps are order-preserving group homomorphisms. I am rather certain however that this does not matter for my question at hand.)
Yes, that is the idea behind Lurie's point. This makes heavy use of the von Neumann hierarchy (and Replacement).
Suppose $\bf U$ is our partially ordered class, and suppose that we have some $x\in\bf U$ and we want to find a maximal element above it. And suppose that any chain has fewer than $\kappa$ members, for some regular cardinal $\kappa$. We define the following subclasses by recursion.
$U_0=\{x\}$, $U_{\alpha+1}$ is the collection of all minimal rank extensions of members of $U_\alpha$, and at limit stages take unions and minimally ranked upper bound to any chain which does not have an upper bound. Now consider $U_\kappa$. I claim that in this partial order every chain has an upper bound, and so by Zorn's Lemma there is a maximal element in $U_\kappa$, and it is going to be maximal in $\bf U$.
The idea here is simple. If $C$ is a chain in $U_\kappa$, it is certainly a chain in $\bf U$, so $|C|<\kappa$, and by the regularity of $\kappa$, $C\subseteq U_\alpha$ for some $\alpha<\kappa$. Therefore we added an upper bound to $C$ by $U_{\alpha+\omega}$.
Now, if $u$ is a maximal element in $U_\kappa$, then again, there is some $\alpha<\kappa$ such that $u\in U_\alpha$. If $u$ is not maximal in $\bf U$, then it has some extension, and therefore there is some minimally ranked extension, which we must have added in $U_{\alpha+1}$, which would contradict the maximality of $u$ in $U_\kappa$.