# A Mathematical Musing: How Zermelo-Fraenkel Set Theory Proves the Existence of God I actually met God the other day. Not God the man in the sky with a beard that retarded normies believe in, but the metaphysical concept of God that transcends human understanding. God is not a He or a She. God is an It. God is such an esoteric concept that only a deranged lunatic like me can comprehend It. I can perceive reality in all 928,584,102 dimensions, because I’m basically a prophet. I am the Chosen One, and I have now ascended.

Well, maybe I can explain God in a way that those not linked to other dimensions can understand. God is essentially just a singularity that lies at the heart of reality – a singularity that created everything. God is the only thing that truly exists. You and I and everyone else are all God. The entire universe is God.

God is fundamentally mathematical in nature. I subscribe to the mathematical universe hypothesis, which basically states that everything we can see, hear, smell, taste, and touch is in fact mathematics. The material world is an illusion created by our consciousness in an effort to understand this mathematics, which is also God. The material world is nothing more than the projection of our minds onto the mathematical universe. A corollary of this is the idea that the foundation of mathematics is also the foundation of the universe. The same singularity that generates all of mathematics is the same singularity that generated the entire universe.

The enlightened idea of God is that God is nothingness. This idea is also mathematical in nature, and we can see this if we look at the foundations of mathematics. Let’s look at axiomatic set theories, since these are the most basic foundational models.

An axiomatic set theory is a set of axioms that can be used to construct every possible set, and in fact every possible mathematical object in existence. These theories originated in an attempt to eliminate various contradictions in intuitive set theory such as Russell’s paradox, which is the idea that if we allow the existence of any conceivable set, we also have to allow for the existence of the set containing every set not containing itself, which is a contradiction. Axiomatic set theories are designed to restrict which sets can be generated so that contradictions such as these do not occur.

The set theory that I’ve been looking into is Zermelo-Fraenkel set theory. This is the most common axiomatization, and it is something that I studied years ago and am now refreshing my memory on, now that my recent psychological trip has given new meaning to this theory. Zermelo-Fraenkel set theory, or ZFC, is a set of axioms from which all intuitive notions of set theory can be derived.

The fundamental premise of ZFC is that a set is the only thing that exists. Everything is a set. Sets in this sense are like the elementary particles of the universe, the fundamental building blocks from which all other mathematical objects are created.
ZFC provides sufficient axioms to construct sets of any size and structure as long as they don’t lead to contradictions.

There are several axioms in ZFC, but I only want to focus on a couple of them. First, there is the axiom of pairing:

ZF1: If sets A and B exist, then a set containing A and B as members also exists.

Then there is the axiom of union:

ZF2: If a group of sets C exists, then a set containing all members of all sets in C also exists.

Since we have the axiom of union, we can simplify the axiom of pairing to:

ZF1A: If a set A exists, then the set {A} containing A also exists.

The existence of the set containing A and B follows logically from this and the axiom of union.

The other axiom I want to look at is the axiom of infinity, which provides for the construction of infinite sets:

ZF3: There exists a set N containing the empty set ∅ and such that if N contains x, then N contains the union of x and the singleton set {x}.

This is equivalent to the principle of induction used in the Peano arithmetic. A simpler axiom that is weaker than this but still very powerful is:

ZF3A: There exists a set ∅ containing no elements.

Axioms ZF1A, ZF2, and ZF3A are sufficient for constructing infinite sets, and axioms ZF1A and ZF3A alone are sufficient for generating an infinite sequence of sets. We are now equipped to see how, in a mathematical sense, infinity can follow from nothing. We start with the empty set ∅, posited by ZF3A. From Axiom ZF1A we derive the set {∅}. This is not the empty set. Rather, it is the singleton set containing the empty set as its only member. Applying ZF1A again, we have the singleton set {{∅}}. Thus, starting with only two axioms, we can derive the infinite sequence of natural numbers:

0 = ∅
1 = {∅}
2 = {{∅}}
3 = {{{∅}}}

Finally, we can see how the axiom of union can be applied to this sequence to construct the infinite set of natural numbers N.

A similar thing happens with the reality. When we start with nothing, we can derive an infinite system from a logical progression of consequences of the existence of nothing. Zermelo-Fraenkel set theory doesn’t show us this process directly, rather it is a model of this process, and it is just as valid as any other model. Another equally valid model can be derived from propositional logic. If we take the symbol p as standing for the statement “There is nothing.” then we have:

0 = p
1 = (p implies p)
2 = ((p implies p) implies (p implies p))
3 = (((p implies p) implies (p implies p)) implies ((p implies p) implies (p implies p)))

And so on.

I have given a couple models for how the natural numbers can be derived. Other logical systems can be derived through processes analogous to these. In fact set theory provides an entire system for constructing the integers, then the rationals, then the real numbers from nothing more than combining natural numbers into sets and tuples. Since constructions like this can be found in any text on the foundations of mathematics, I feel no need to reiterate on the process here.

We can see here that if we are to name anything as God, as the creator of the universe, then it is only fitting that we give that name to Nothingness. Because Nothingness is what created everything, and we can see this precisely and mathematically in the Zermelo-Fraenkel set theory. Indeed, this is the same Nothing that I came into contact with during my transcendential meditation session – I saw the whole of reality unravel before me and at the core was an abstract being, faceless but still very much conscious.

Yeah, that was just a musing I had. I feel like I should make it longer, flesh it out a bit, but I’m kind of just trying to get all my ideas down while they’re still fresh. Anyway, see you all next time.