Is there a non-standard set theory that makes use of a null element?

Has there ever been something like an "empty element" or a "zero entity" been proposed, or at least is there any more or less standardized symbol to denote such a (no)thing?

E.g., if $\epsilon$ denoted what I am interested in, it would be that $\{0,1,\epsilon\} = \{0,1\}$ and $\{\epsilon\} = \emptyset$.

I am aware that this would cause unpleasant side effects, as such a $\epsilon$ would be an element of the empty set which by defintion doesn't have any elements; whether introducing an explicit notation for a non-existent entity is useful is surely questionable; and I don't plan to revolutionize set theory with a whole new type of entity either.
I'd just like to know:

1. Has something like this ever been proposed in logic? How would this idea be formalized, and what effects would the assumption of a null element have w.r.t. essential axioms of standard set theory? Can the notion of a null element in some way be brought into accordance with basic set theoretic assumptions, possibly even be of avail for one or the other problem, or would it have such illogical consequences that an empty element is unjustifyable even in alternate set theories?

2. As for the practical aspect, apart from the philosophical point of view, is there any notational convention/symbol that roughly resembles what I described (something in the style of $\epsilon$)?
The closest I could find is $\uparrow$ being used for undefinedness, but that still doesn't quite fit into context here. If the answer is "Such a notation doesn't exist because it has never been used ", that's fine too.

Edit: Originally I was really just interested in notation, but the question has attracted such good theoretical comments that I reworded my question so as to elaborate on the first part more explicitely.

• This could be the start of a grand expanded theory of sets containing elements which do not contribute to cardinality.. or not. – Ravi Dec 9 '16 at 22:39
• I am certain no one has done this as it has no clear purpose and would require changing the underlying logic. If the empty set exists, and existential generalization is a thing, then you get a contradiction immediately from the simple statement $\epsilon\in\varnothing$. – Malice Vidrine Dec 9 '16 at 22:57
• A math problem clearly inspired by [Antigonish](en.wikipedia.org/wiki/Antigonish_(poem). – hardmath Dec 10 '16 at 1:53
• @Andrés E. Caicedo How is a question about the elements of a set and and its meaning w.r.t. set theoretic assumptions not about the theory of sets? Whatever definition you seem to presuppose is not what the tag wiki says; otherwise please explain me what your definition is, because I really don't see what the problem is. – lemontree Dec 10 '16 at 9:23
• @AndrésE.Caicedo: your claim that "Set Theory has a well established meaning that does not apply here" is arrant and arrogant nonsense. Your reaction is a disincentive to those who wish to ask intelligent questions about the formalisation of set theory. I have put the tag back and will be voting to reopen the question if and when it is closed. – Rob Arthan Dec 10 '16 at 23:42

1 Answer

It's funny I had this same exact idea like a week ago.

The bigger problem I saw is that first of all if you include such an element in the "common" set theory, you pretty much just have a cool way to describe the empty set: everything should work quite nicely but with not much effective difference. For example two disjoint sets $A$ and $B$ intersect on $A\cap B=\{\epsilon\}$, since both they contain $\epsilon$, but there is no real difference from $A\cap B=\varnothing$.

If however you add this element in order to "extend" set theory slightly, quite a number of other problems arises.

First of all, we want to define this $\epsilon$, and the main property we want for this element is to be contained in every available set: $$\epsilon\in S\quad \forall S \textit{ "set"}$$

However, if you require this (and some form of axiom of specification), you get that, for any given suitable property $P$: $$\epsilon\in\{a\mid P(a)\text{ is true}\}\qquad\text{and}\qquad\epsilon\in\{a\mid \neg P(a)\text{ is true}\}$$

which means that $\textbf{any proposition or property is both true and false for such an element}!$

Even the "always true" and "always false" ones, or the property "$x\in x$". We in fact have that both $\epsilon\in\epsilon$ and $\epsilon\notin\epsilon$ hold at the same time. But even more normal properties give trouble: for example $\epsilon\in\mathbb{N}$ and $\epsilon\notin\mathbb{N}$. It doesn't seem something easy to fix at all.

In conclusion, my final thought is that either you do substantially nothing aside of a change of quite a bit of notations, or alternatively you are required to fix a huge quantity of problems that this "nonelement" will create.

• Also, the proof of Cantor's theorem wouldn't hold! If we consider $f(x) = \{x\}$, $A = \{x | x \in f(x)\}$ would be impossible to build, because even the null set would be mapped to by $\epsilon$. – AmagicalFishy Jan 4 '18 at 22:24