In axiomatic set theory every axiom uses the existence of some set as a prerequisite for the formula of the axiom. For example the axiom schema of specification which looks like:
The axiom is initiated by mentioning the quantifiers (∀x) and (∃y) which are read “for all x” and “there exists a y”, respectively. In this article I focus on this “there exists” property. The vagueness of that quantifier is the subject of this entire article.
What, exactly, does the existence quantifier indicate?
A new axiom for set instantiation
With the term ‘set instantiation’ I mean the act of giving a set an identifier, such as A, so that it can be referred to. I propose that the existence quantifier ∃ have been assumed to be legal to use anywhere, due to a naive and faulty assumption of the requirements for set instantiation. I propose that one cannot simply conjure up the existence of any set A without stating the proper circumstances, if A is to be used as a identifier in a formula involving more than A itself. For example, the statement
A ∨ A
requires no existence qualifier since it only involves the set A. But for the standard-set-theory statement
A ∨ B
to even make sense at all we need to explicitly specify that each of the two sets A and B exist in the same manner of existing! I escape the problem of defining exactly what existence means by simply stating that they all must use the same manner of existing. If they have different manners of existing then the statement is complete nonsense. Thus such a statement is only legal if we accompany it with the assertion that all of them use the same manner of existing:
((A,B) ⊆ S)(A ∧ B)
Where S is the manner of existing and A and B are members of it. It simply means there is some greater set S which both of them are members of. This is a quite significant addition. Indeed I insist that for any set operations of any kind to make intuitive sense, all of the variables included in the operation must belong to the same greater set S. That, I propose, is a new axiom schema that I state formally as follows:
φ(A1,A2…,An) ⇔ (φ, A1,A2…,An) ⊆ S
It can be verbally stated as: “for every expression φ using parameters, there must be a greater set S that all those parameters, as well as φ itself, must belong to”.
Oh my, that escalates quickly…
You quickly realize that this axiom schema implies something about each and every logical statement that you can make. Therefore, it implies something about all axioms. Therefore this new axiom, by logical extension, steadily positions itself as mightier than all other axioms!
It says that all logical statements that exist are children to a parent logical statement that also exists. This neatly arranges not only all of mathematics, but all of existence into a tree-like structure. Said in another way, it means that if that axiom is indeed an axiom then all of reality is a tree-like structure. Not sortof, not hypothetically, not abstractly, but really. I will refer to this axiom as the tree axiom.
At the root of the tree
The tree axiom indicates that all of existence is arranged into a tree-like structure. But if the nature of existence is shaped like a tree, one of the first questions we ask is naturally: what would then be at the root of the tree? There must be something there that somehow doesn’t have a parent, despite the axiom!
We should follow our intuition which says that as we approach the root of the tree, the number of available mathematical symbols shrink. Somewhere near the root of the tree we have reached the point where we no longer have both the symbols A and B; only the symbol A remain. What kinds of statements can one make with only a single symbol? One attempt would be
A ∨ A ∧ ¬A ∧ A
which we can clearly see doesn’t come close to only being a single symbol, since aside from A it also uses the symbols ∨, ∧ and ¬. These extra symbols represent logic operations that are concepts that somehow must also disappear near the root of the tree. We can get a slight bit closer to the root by applying the tree axiom which implies that even closer to the root we should find the statement:
(A ⊆ S)
which in turn is a logical statement, so that:
((A, S) ⊆ S)
Which is a dead-end. We can get no further that way. But how do we then figure out what the heck the universal S is? What is this foundation that all other statements rely upon?
Clearly, for S to require no further parent, the top statement must include nothing but S, an attempt at which could look like this:
(S ⊆ S)
But that includes the relationship indicator ⊆, which at this foundational level cannot exist and must have somehow been replaced by… S:
(S S S)
Which by making the character in the middle bold I attempt to show you that the set S tries to perform “operation itself” on itself. We have arrived at a guess of what the most foundational…. thing… is in the universe. We are not quite there yet, but now we are getting somewhere, and we have developed an intuition of what kind of thing we are looking for. We just need to arrive at something that looks like (S S S) by some kind of rigorous method.
Solving the new inconsistency
What I am talking about here is, as any young set theorist would immediately point out, a kind of universal set. Such a thing is ruled out by Zermelo-Fränkel set theory, by the will of the previously mentioned Axiom Schema Of Specification. Thus I cast out that axiom. I dislike it because it doesn’t fit my intuition. But doing so I allow in the inconsistencies again, and my theory makes no sense again. So I append a new “axiom” in its place, to solve the inconsistency. But it isn’t as simple as simply writing a new statement of set logic with mathematical symbols, because I need to say something new about set theory as a whole, something that changes the very nature of what a set is. By doing so I would have to introduce a bunch of new mathematical symbols that would just confuse you, the reader. So instead I will explain it in words, and perhaps sum it up in the end with the new symbols.
This will be the topic of the next article in this series.