God and the Axiom of Choice

"If they will not believe Moses and the Prophets, neither will they believe if someone should rise from the dead."

All conflicts are fundamentally religious conflicts, as can be seen from asking a toddler. If you wander far enough down the trail of repeated "why", you eventually get to a claim that can only be answered with "that's just how the world is." That answer is a religious answer, an answer based upon authority, be it only the authority of your own senses.

This is deeply troubling to anyone who thinks about it for too long, for we either crave complete relativity, or an objective truth that we know for certain. This middle ground where there are things outside ourselves, that we cannot dictate, but which nevertheless we cannot know (and thus cannot control) drives us mad.

This madness takes three forms, of excess and two styles of deficit. The excess is that of a certain type of so called "religious" person, whereby we credulously think we know that which we do not, and claim that the categories of our answers are the only way we can analyze the problem. The first deficit is that of the naturalist, whereby we deny that anything can be known that cannot be demonstrated to the senses. The second deficit is that of the relativist, whereby even our senses are claimed to be arbitrary and only will determines truth. That is, the faults are to say "I know all about God¹," "I know there is no God," and "I know there is a God, and it’s me."

"But," you say, "what other option is there? Either there is God or there is not." Certainly, but just because in abstraction either P or not P is true does not mean we can know which one it is. Nor is this a unique phenomenon. Mathematics proffers examples, less emotionally fraught, which can demonstrate the principle. I speak of the axiom of choice.

Mathematics is built upon a series of 8 axioms called the Zermelo-Fraenkl axioms of set theory. Most of them are very, very basic, almost definitional, things, such as "two sets are equal if they contain all the same elements" or "there is a set that contains all the elements that a group of sets contain," and from these basic statements, mathematicians can, and have, built all of mathematics. Well, that isn't entirely true. From those 8 statements, mathematicians have built most of mathematics. There is one more axiom, much less clear, needed to get all of it.

This axiom is called the axiom of choice. It states, very roughly, that if you have an infinite collection of identical objects, there is a way to choose a unique element from it. The common analogy is a pile of identical socks. If the pile is actually infinite², you need the axiom of choice to match each sock with a partner. This seems benign, but it turns out to have some highly irregular and counterintuitive consequences if assumed³. Unfortunately, it is also logically necessary⁴ to some of the most useful results in Mathematics, undergirding most of modern mathematics, and therefore most of modern technology.⁵

This situation of both being needed to justify the things we really hope are true and forcing us to accept things we really don't want to be true reminds me of the arguments over God in the wider world, and I claim the parallels continue.

Firstly, the axiom of choice doesn't introduce a contradiction. It has been proven by Kurt Godel⁶ that any contradiction in the Zermelo-Fraenkl axioms was already there before assuming the axiom of choice. Similarly, the Doctor writes⁷ “everything we see in the world can be accounted for by other principles, supposing God did not exist. For all natural things can be reduced to one principle which is nature; and all voluntary things can be reduced to one principle which is human reason, or will.”⁸ But once you translate Thomism into math, the two statements say the same thing. Neither the axiom of choice, nor God, introduce a contradiction into the system in question.

Secondly, the axiom of choice is actually an axiom. Paul Cohen⁹ proved that there is no way to prove the axiom of choice from the other Zermelo-Fraenkl axioms. Similarly, we can't prove God exists in a way that is absolutely beyond doubt¹⁰. It is possible to imagine a logically stable universe in which there is no God. It just happens that as a matter of fact, the one in which we live is not that one, as can be seen by observation, for "the heavens declare the glory of God."

Finally, just as you are free not to assume the axiom of choice, so long as you are willing to dispense with all the results that require it, so too can you deny the existence of God, but only if you are willing to deny all the experiences in the world that only make sense if there is a God. Of course, these are things like "I have inherent worth," "other people matter," "There is a moral law besides self-interest," "'Good,' 'True,' and 'Beautiful' are words that mean something objective," "Justice is not just the will of the stronger," "My actions are not completely deterministic," and so on,¹¹ so although you are welcome to deny them, it will be rather difficult for you to live in a logically consistent way that is also "authentically human." It will work out much better to say "I believe in God, and He is not me.” So, although you are free to do as you will, “as for me and my house, we will serve the Lord.”

1. If you have an allergy to the word "God", then replace it with "Governing Will of all that exists" and that will be good enough for now.

2. in the mathematical sense, not the theological sense.

3. The Banach Tarski paradox is the archetypal example.

4. that is, assuming the axiom of choice, you can prove the result, and assuming the axiom of choice is false, you cannot prove it.

5. The prototypical examples are Zorn's Lemma and the Well Ordering Principle, which are equivalent to the axiom of choice, but other, user facing, theorems requiring the axiom of choice are the Hahn Banach theorems, Tychonoff's Theorem, Baire Category Theorem, Krein-Milman theorem, existence of vector space bases, existence of Hilbert space orthonormal bases, and so on.

6. in book length, but a sketch is here.

7. In the second objection to the question “does God exist,” the first objection being the problem of evil.

8. The reply to this objection, for posterity, is: “Since nature works for a determinate end under the direction of a higher agent, whatever is done by nature must needs be traced back to God, as to its first cause. So also whatever is done voluntarily must also be traced back to some higher cause other than human reason or will, since these can change or fail; for all things that are changeable and capable of defect must be traced back to an immovable and self-necessary first principle, as was shown in the body of the Article.”

9. in this paper.

10. In Thomistic language, the existence of God cannot be demonstrated a priori, although it can be observed from its effects.

11. Defense in depth that each of these claims is actually equivalent to the existence of God is beyond the scope of this essay.