## A Bad Mathematical Refutation of Atheism

At some point a few months ago, someone (sadly I lost their name and email) sent me a link to yet another Cantor crank. At the time, I didn't feel like writing another Cantor crankery post, so I put it aside. Now, having lost it, I was using Google to try to find the crank in question. I didn't, but I found something really quite remarkably idiotic.

(As a quick side-comment, my queue of bad-math-crankery is, sadly, empty. If you've got any links to something yummy, please shoot it to me at markcc@gmail.com.)

The item in question is this beauty. It's short, so I'll quote the whole beast.

MYTH: Cantor's Set Theorem disproves divine omniscience

God is omniscient in the sense that He knows all that is not impossible to know. God knows Himself, He knows and does, knows every creature ideally, knows evil, knows changing things, and knows all possibilites. His knowledge allows free will.

Cantor's set theorem is often used to argue against the possibility of divine omniscience and therefore against the existence of God. It can be stated:

1. If God exists, then God is omniscient.
2. If God is omniscient, then, by definition, God knows the set of all truths.
3. If Cantor's theorem is true, then there is no set of all truths.
4. But Cantor’s theorem is true.
5. Therefore, God does not exist.

However, this argument is false. The non-existence of a set of all truths does not entail that it is impossible for God to know all truths. The consistency of a plausible theistic position can be established relative to a widely accepted understanding of the standard model of Cantorian set theorem. The metaphysical Cantorian premises imply that Cantor’s theorem is inapplicable to the things that God knows. A set of all truths, if it exists, must be non-Cantorian.

The attempted disproof of God’s omniscience is, from a meta-mathematical standpoint, is inadequate to the extent that it doesn't explain well-known mathematical contexts in which Cantor’s theorem is invalid. The "disproof" doesn't acknowledge standard meta-mathematical conceptions that can analogically be used to establish the relative consistency of certain theistic positions. The metaphysical assertions concerning a set of all truths in the atheistic argument above imply that Cantor’s theorem is inapplicable to a set of all truths.

This is an absolute masterwork of crankery! It's remarkably silly argument on so many levels.

1. The first problem is just figuring out what the heck he's talking about! When you say "Cantor's theorem", what I think of is one of Cantor's actual theorems: "For any set S, the powerset of S is larger than S." But that is clearly not what he's referring to. I did a bit of searching to make sure that this wasn't my error, but I can't find anything else called Cantor's theorem.
2. So what the heck does he mean by "Cantor's set theorem"? From his text, it appears to be a statement something like: "there is no set of all truths". The closest actual mathematical statement that I can come up with to match that is Gödel's incompleteness theorem. If that's what he means, then he's messed it up pretty badly. The closest I can come to stating incompleteness informally is: "In any formal mathematical system that's powerful enough to express Peano arithmetic, there will be statements that are true, but which cannot be proven". It's long, complex, not particularly intuitive, and it's still not a particularly good statement of incompleteness.

Incompleteness is a difficult concept, and as I've written about before, it's almost impossible to state incompleteness in an informal way. When you try to do that, it's inevitable that you're going to miss some of its subtleties. When you try to take an informal statement of incompleteness, and reason from it, the results are pretty much guaranteed to be garbage - as he's done. He's using a mis-statement of incompleteness,and trying to reason from it. It doesn't matter what he says: he's trying to show how "Cantor's set theorem" doesn't disprove his notion of theism. Whether it does or not doesn't matter: for any statement X, no matter what X is, you can't prove that "Cantor's set theorem" or Gödel's incompleteness theorem, or anything else disproves X if you're arguing against something that isn't X.

3. Ignoring his mis-identification of the supposed theorem, the way that he stated it is actually meaningless. When we talk about sets, we're using the word set in the sense of either ZFC or NBG set theory. Mathematical set theory defines what a set is, using first order predicate logic. His version of "Cantor's set theorem" talks about a set which cannot be a set!

He wants to create a set of truths. In set theory terms, that's something you'd define with the axiom of specification: you'd use a predicate ranging over your objects to select the ones in the set. What's your predicate? Truth. At best, that's going to be a second-order predicate. You can't form sets using second-order predicates! The entire idea of "the set of truths" isn't something that can be expressed in set theory.

4. Let's ignore the problems with his "Cantor's theorem" for the moment. Let's pretend that the "set of all truths" was well-defined and meaningful. How does his argument stand up? It doesn't: it's a terrible argument. It's ultimately nothing more than "Because I say so!" hidden behind a collection of impressive-sounding words. The argument, ultimately, is that the set of all truths as understood in set theory isn't the same thing as the set of all truths in theology (because he says that they're different), therefore you can't use a statement about the set of all truths from set theory to talk about the set of all truths in theology.
5. I've saved what I think is the worst for last. The entire thing is a strawman. As a religious science blogger, I get almost as much mail from atheists trying to convince me that my religion is wrong as I do from Christians trying to convert me. After doing this blogging thing for six years, I'm pretty sure that I've been pestered with every argument, both pro- and anti-theistic that you'll find anywhere. But I've never actually seen this argument used anywhere except in articles like this one, which purport to show why it's wrong. The entire argument being refuted is a total fake: no one actually argues that you should be an atheist using this piece of crap. It only exists in the minds of crusading religious folk who prop it up and then knock it down to show how smart they supposedly are, and how stupid the dirty rotten atheists are.