# 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.

• You can't use mathjax here. – curiousdannii Sep 27 '18 at 13:34
• @curiousdannii This is why I put "mathjax here" on top. – Jack Bauer Sep 27 '18 at 13:36
• Well you should edit it to make it readable. You can use <sub> and <sup>, and several (probably most) of the symbols are in unicode. – curiousdannii Sep 27 '18 at 13:43
• Well the maximal set of a powerset if of course the original set itself. That is unique. What does it have to do with axiom of choice or God? I understand the last part tries to say that an all-mighty God implies axiom of choice, but doesn't that basically just mean axiom of choice is assumed? – kutschkem Oct 1 '18 at 14:18

2. As you said we have God $\supseteq$ God $\cup$ ${$existence$}$ as existence is a property, God $\cup$ ${$existence$}$ is greater or equal than God, but God is maximal (see 1).