Why does the axiom of choice imply the existence of a unique God?

The following is paraphrased from *A Passion for Mathematics* by Clifford A. Pickover (read in Kiefer Sutherland's voice): (mathjax here ) **Theorem**: The axiom of choice is equivalent to the existence of a unique God (St. Anselm, Aquinas, and others). > Proof: > > First, suppose the axiom of choice. Partially order the set of subsets > of the set of all properties of objects by inclusion. This set has > maximal elements. God is by definition (due to Anselm) a maximal > element set. > > We prove existence: God $\subseteq$ God $\cup$ $\{$existence$\}$, so > God $=$ God $\cup$ $\{$existence$\}$. Therefore, God exists. > > We prove uniqueness: Let God and God′ be two gods, then God $\cup$ > God′ $\supseteq$ God (due to Aquinas) $\implies$ God $\cup$ God′ = God > $\implies$ God $\subseteq$ God′ Similarly, God′ $\subseteq$ > God. Therefore, God is unique. > > Second, suppose the existence of a unique God, omnipotent, omniscient > and amoral (or omnibenevolent. It does not matter for this context). > Given an index set $R$ and collection of sets > $\{A_{\alpha}\}_{\{\alpha \in R\}}$, pray that the unique God picks, > by omnipotence, $x_{\alpha} \in A_{\alpha}$ for each $\alpha \in A$. > Then $$\{x_{\alpha}\}_{\alpha \in R} \in \prod_{\alpha \in R} A_{\alpha}$$ as required. Questions: 1. What is a "maximal element set"? I could not find this online. I do not remembering learning this in philosophy of religion class when I was in bachelor's. 2. In existence, how does God $\subseteq$ God $\cup$ $\{$existence$\}$ imply God $=$ God $\cup$ $\{$existence$\}$? I guess that we have somehow already had God $\supseteq$ God $\cup$ $\{$existence$\}$. 3. In uniqueness, why do we have have that God $\subseteq$ God′ instead of God $\supseteq$ God′? 4. Is uniqueness required for the second direction? I can think of only the unique choice of $x_{\alpha}$ or something of the sort. I think I am fine with the second direction. Update: I think [this link](http://ion.uwinnipeg.ca/~currie/logic.pdf) might have some answers.