3

The title seems quite bit more generalized, but I'm saying about the philosophers and mathematicians in the past who discussed about the concept of *nothing*, or the *empty set*. I'm currently studying set theory again, and I'm trying hard to understand concepts as total as possible. But I don't understand that why have we had a problem about understanding the concept of the *empty* set, and why there is an axiom called "The axiom of existence", which also called "The axiom of empty set".

First, let me start with introducing my understanding of *nothing*;
Before define it, I have to introduce a *sub-universe* (I haven't found any rigorous way to explain it yet, but only informal way.).

Definition 1.Asub-universeis a part of my(or our) perceptions.

The reason why I define this is to avoid some paradoxes(e.g. Russell's paradox) from *The universe of discourse*. In mathematics, we limit our perceptions or analysis by means of the axioms. The sub-universe has similar contexts, but we can pick smaller universe whatever we want if we adopt this concept. (be cautious that this concept can only be applied relevantly when the rules of inference or rules of thoughts satisfied. i.e., it must not contain any paradox.)

Now I introduce the definition of *nothing*;

Definition 2.Let S denotes a sub-universe. We say a thing isnothing in Sif and only if the thing dose not contains any element from it'ssub-universe.

For example, assume that there is a box which contains a blue ball and a green ball. And then let us denote this as a set B, B={blue ball, green ball}. And define a sub-universe, S, in this situation which contains the set B. And now, I will remove 'the balls' from the box, which can be denoted by {}. Then the box is *nothing* in S.

From this view, if we can expand a sub-universe as total as possible, then I think that there is no problem with the concept of *nothing*. I certainly have no problem in understanding *nothing*, and so I really can't get any necessity on "The Axiom of existence". To sum up, my question is simply;

Question.Why "The axiom of existence"?

2It's my sense that the people who have a "problem" with the empty set are numerous amateur philosophers on Internet discussion fora. By "amateur philosopher" I mean everyone with enough higher brain function to register a handle on a forum and bang on a keyboard. I include myself in that category. Actual professional philosophers understand that the empty set is like an empty bag of groceries. It doesn't have anything in it. Which means that it contains all the purple unicorns. Everyone understands this. Are there reputable philosophers who "don't understand the concept" of the empty set? – user4894 – 2014-07-24T20:04:28.310

Related to my question earlier: http://philosophy.stackexchange.com/questions/9246/isnt-it-absurd-to-suppose-that-sets-can-be-empty-or-can-contain-other-sets?rq=1 I still find the concept troubling if you try to interpret it at all. But if you see it as just a calculus, which I suspect is what most mathematicians actually do, then you just "turn" off that part of the intellect that wants to see it as anything but.

– Kevin Holmes – 2014-07-24T20:45:53.4001If you are defining things in terms of your perceptions, you are not doing set theory. – WillO – 2014-07-24T21:32:48.427

@user4894 Of course professional philosophers understand the mathematically relevant properties of the empty set. However there are metaphysical properties of sets (and so of the empty set) that are puzzling: Having the members some set A actually has is sufficient for being A. It is highly controversial if

concretalike tables have properties that are sufficient for their identity. But without doubt sets have such properties. Why is that so? This issue has been discussed by Graeme Forbes and others. – sequitur – 2014-07-24T21:45:47.9801About the specific question regarding the

null-set axiom(the "existential" axiom regarding the empty set) we need itinto axiomatic set theorysimply because we are working into a mathematical environment and not in a philosophical one. In a math theory of sets there are no concepts of "nothing", but only two "concepts" :set, i.e. every "object" in the "universe" of set theory is a set, andmembership, i.e. there is only one "relevant" relationship between two objects of this universe : the membership relation. Nothing more. – Mauro ALLEGRANZA – 2014-07-25T07:05:19.303