# A Mathematical Musing: How Zermelo-Fraenkel Set Theory Proves the Existence of God

I’ve recently been exploring a lot of metaphysical and enlightenment concepts through meditation. I have been meditating for several years now, and have become so good at it by this point that I am able to enter a heightened state of consciousness and come into direct contact with God. Keep in mind that when I say “God” I’m not talking about a bearded man in the sky. Rather, I refer to the metaphysical concept of God, which is a universal consciousness, a concept of Being, a singularity that created everything. God is the nothingness that exists at the heart of reality, and all time, space, matter, and energy springs from this nothingness. I believe that enlightenment gives us insight into how this process occurs.

At the same time that I have been doing this enlightenment work, I have been exploring another discipline that parallels and corroborates this understanding of creation – the axiomatic formulation of set theory and its use as a basis for all of mathematics. I am a believer in the mathematical universe hypothesis. Everything we can see, hear, smell, taste, and touch is in fact mathematics, but our consciousness interprets it as material through our senses. The material world is an illusion created by the projection of our mind onto the mathematical universe. A corollary of this is the idea that the foundation of mathematics is also the foundation of reality. The same singularity that generates all of mathematics is also what has generated the entire universe.

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 as an attempt to eliminate various contradictions in intuitive set theory such as Russell’s paradox, which is that if we allow the existence of any conceivable set, we also have to allow the existence of the set that contains 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 meditation journey has given new meaning to this theory. Zermelo-Fraenkel set theory, or ZFC, is a set of axioms from which all the 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 Ø 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 sufficient for producing infinite sets is:

ZF3A: There exists a set Ø 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 Ø, posited by axiom ZF3A. From axiom ZF1A we derive the set {Ø}. 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 {{Ø}}. Thus, starting with only two axioms, we can derive the infinite sequence of natural numbers:

0 = Ø
1 = {Ø}
2 = {{Ø}}
3 = {{{Ø}}}

Finally, we can easily 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 in metaphysics. 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 does not 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 is derived from propositional logic. If we take the symbol p as standing for the statement “Nothing exists.” 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 to 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 we come into contact with when we attain enlightenment through meditation, and when we experience this Nothing firsthand, we find all the qualities traditionally attributed to God: infinite love, infinite power, infinite wisdom, etc. This is the ultimate truth.

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 when they’re still fresh. Anyway, see you all next time.