Alternative set theories?

Question

Is there a version of set theory that allows the existence of a set that does not admit the empty set as a member? I.e., reject the axiom $A\cup \emptyset = A$

Answer 1

Consider a new relation $\mathbin{\in'}$ among sets, defined by $x\mathbin{\in'}y$ iff $x\in y$ OR ($y$ is empty AND $x=y$). Note that the original relation $\in$ can be recovered from this $\mathbin{\in'}$ by: $x\in y$ iff $x\mathbin{\in'}y$ AND $y\mathbin{\not\in'}y$ (this uses the axiom of Foundation, which guarantees in particular that $y\not\in y$).

If we interpret this relation $\mathbin{\in'}$ as a belonging relation among sets, then there is no set which is empty-in-the-$\mathbin{\in'}$-sense: it has been replaced by an atom $o$ which satisfies $o\mathbin{\in'}o$ (and it is the unique set with this property).

Now we can axiomatize set theory for the $\mathbin{\in'}$ relation: indeed, take the axiomatization of ZFC and replace every occurrence of "$x\in y$" by "$x\mathbin{\in'}y$ AND $y\mathbin{\not\in'}y$", and add two further axioms which say that (1) for all $x,y$, if $y\mathbin{\in'}y$ and $x\mathbin{\in'}y$ then $x=y$, and (2) if $y$ is such that there is no $z\neq y$ with $z\mathbin{\in'}y$, then in fact $y\mathbin{\in'}y$ (equivalently, these axioms can be formulated by saying that "$x\mathbin{\in'}y$ iff $x\in y$ OR ((there is no $z$ such that $z \in y$) AND $x=y$)" where $\in$ has been replaced in the right hand side of the "iff" in the same manner as in the axioms of ZFC; this is logically equivalent to (1) and (2)). It is then easy to see that the original ZFC set theory can be recovered from this ZFC′ by defining $x\in y$ iff $x\mathbin{\in'}y$ AND $y\mathbin{\not\in'}y$. So both theories are equiconsistent (they are co-interpretable).

So yes, this provides an "alternative set theory" in which there is no such thing as the $\mathbin{\in'}$-empty set: instead, the axioms guarantee the existence of $o$ such that $o$ is the unique $\mathbin{\in'}$-element of $o$. (Furthermore, even if we interpret $\mathbin{\in'}$ as $\in$, I think the axioms of ZFC′ minus Foundation′ and the axiom (2) are consistent with ZFC minus Foundation, or something like that. So it's not even so wildly "alternative".)

Of course, this set theory is profoundly uninteresting. It just stupidly replaces the empty set by a self-containing atom by changing the $\in$ relation. But it can be done.