Yes GiC, I *know*. That's the point. You have defined a set as being a set of things in a particular state, but that particular state cannot be a state. Non-existence cannot be the state of some*thing* because it is the state of no*thing*, i.e.,non-existence is not the state of any*thing*.

And yet you speak of it as the negation of existence, philosophers have spoken of nothing or non-existence as the negation of existence for thousands of years. Why do you believe we can denote it linguistically (which we obviously can if we can speak of evil as non-existence), but not mathematically which is far more fundamental than language. The very act of arguing that something cannot be represented mathematically is the creation of a contradiction in said argument.

Yes, GiC, that's what I'm saying. The cardinality of the empty set precludes it from being "the set of non-existent things" because non-existence, by definition, has no existence. To speak of non-existence as *something which exists* (in your case, a "set with no elements") is self-contradictory.

It is not self-contradictory, infact that the empty set, which is (in theory, at least) existent, has no elements is essential to the existence and consistency of the natural numbers.

Such a proposition is meaningless and nonsensical because you cannot negate the concept of existence and yet speak of its (the negation’s) existence in the same proposition. The empty set cannot be equated with "the set of non-existent things" because "the set of non-existent things" cannot, by definition, exist, but the empty set *does* exist, therefore the empty set cannot be "the set of non-existent thiings".

Why do you belive set theory to be restricted merely to the existent? Set theory often deals with things, the existence of which is theoretically impossible from theoretical computers to theoretical shapes, which can never be constructed or drawn. Relative to our universe, they are non-existent, and will never be existent, but we talk about them and deal with them on a daily basis in mathematics. Yet I can, and have, taken the set of all computers that do not exist (it's simply the complement of the set of computers that do exist, which are turing equivalent and simpler machines, it's rather trivial to define) and worked with this set, these computers have many interesting results even though they simply do not exist. That's the wonderful thing about theory, it's not confined to existence, and set theory even goes beyond this. Sets can contain those things which do exist and do not exist, and other sets can contain sets that contain these sets. Set theory is beyond existence, infact it is infinitely beyond existence. If I can define something linguistically, I can make a set out of it, and even things I cannot define linguistically (nor, ultimately, really comprehend, as my mind is merely a turing machine) or even define the results of linguistically I can still put into sets (which is often how computers beyond turing machines are dealt with, we put the machine and its results in a set, then try to figure out what we can and what our small minds can understand about this machine through the use of the theory of computation which is heavily dependent on set theory). The existent and the non-existent are everyday topics in mathematics, and what one learns from mathematics, when one begins to deal with these things, is that they're not all that different from each other and both are studied in similar manners.

The empty set is not *nothing* it is *something*, it has existence, whereas non-existence is *nothing* and has no existence. There cannot be anything “outside” existence because any such existent would then become part of reality, i.e., existence, but if you equate non-existence with the empty set, you imagine non-existence as some "void" contained in an existing set, but the instant you do that, you are not describing non-existence, because you are describing an existent. Your conclusion that non-existence is contained within existence is the classic "fallacy of the reification of the zero" because you concretely have included non-existence in the set of existing things.

I am still confused why you are so obsessed with the empty set, I never defined non-existence as they empty set; I simply defined it as the complement of the set {x: x exists}, who ever said it has to be the empty set?