Past exam of the logic and set theory course of the University of Oxford / 2007 / B / 8 / a / Solution

Zorn's lemma states that a nonempty partially ordered set in which every chain has an upper bound contains a maximal element. Here a chain means a subset totally ordered by the inherited order; it need not have a greatest member.

View every partial function as its graph, a subset of $\mathbb{R}\times \mathbb{R}$. The family $\mathcal{F}$ is a set by axiom of separation from the power set of this product. Let $\mathcal{C}\subseteq \mathcal{F}$ be a chain and put $h=\bigcup \mathcal{C}$. If two members of the union assign values to the same input, the chain property makes one graph contain the other, so those values agree. Thus $h$ is a partial function.

If $x\in \mathrm{dom}h$ and $q\in \mathbb{Q}$, choose a chain member containing $x$ in its domain. Its rational-scalar closure gives $qx\in \mathrm{dom}h$. If $x,y\in \mathrm{dom}h$, choose two chain members containing them; the larger one contains both, and therefore contains $x+y$ with value $h(x)+h(y)$. Hence $h$ satisfies the required additive identity and domain closure. Consequentlyand it is an upper bound for the chain. For the empty chain its union is the empty function, which satisfies the defining logical implications vacuously, so this case is covered too.