# Trouble comprehending the empty set

## Main Question or Discussion Point

What exactly is a set with no elements? What does it mean? Aren't sets entirely defined by their contents? In what manner is a set with no contents defined?

I'm not arguing for or against such an axiom... I simply want to know what the axiom actually is saying. If an empty set is not well-defined, what is it actually asserting?

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The empty set contains the elements that are the intersection of two mutually exclusive events.

Hurkyl
Staff Emeritus
Gold Member
What exactly is a set with no elements?
Just what you said. If A is an empty set, then, for any x, x is not in A.

What does it mean?
What do you mean by 'mean'? i.e. what sort of semantic interpretation are you using?

Aren't sets entirely defined by their contents?
Two sets with the same membership relation are the same set, if that's what you mean.

In what manner is a set with no contents defined?
For example, by the condition that for any x, x is not a member of that set.

I simply want to know what the axiom actually is saying.
It's asserting that an empty set exists.

CRGreathouse
Homework Helper
What exactly is a set with no elements? What does it mean? Aren't sets entirely defined by their contents? In what manner is a set with no contents defined?

I'm not arguing for or against such an axiom... I simply want to know what the axiom actually is saying. If an empty set is not well-defined, what is it actually asserting?
The empty set is the set of McLaren F1s I own. The empty set is the intersection of the set of red things and the set of nonred things.

Sets are defined on the basis of their contents, which means that:
* The empty set makes sense (it is the set which has no elements)
* The empty set is unique (if there were two empty sets, they would contain the same things [nothing] and so be equal)

A set is indeed defined entirely by its elements. Which is why the empty set is unique.

I prefer to say that the empty set is unique... up to equality.

After all, there is nothing in set theory to say that there is one and only one of each set. It's just that if there were multiple copies of a set, nothing in set theory could at all tell them apart.

I could potentially have a model of set theory where there are 2 empty sets but they are treated identically. Then most sets would have many equal copies. And I could only tell them apart if I had some way outside of set theory that could distinguish them.

CRGreathouse
Homework Helper
Ontologically, it's usual to treat equality as more than just an equivalence relation, but certainly nothing bad would happen if you used an equivalence relation over sets where things that are equivalent 'act the same way' set-theoretically.

Running on empty?

"What exactly is a set with no elements? What does it mean? Aren't sets entirely defined by their contents? In what manner is a set with no contents defined?"

A more tangible speculative physical model might be to take a universe and toss away quanta, and patches of manifold. Supposedly nothing left, it might seem.

However, I'm still having trouble. This is really hard for me, and I feel dumb for failing to perceive something so simple. Here's my next question that may put the issue to rest, for me.

What would be a good definition for empty set, that relies in no way on a notion of a non-empty set, or a definition of a non-empty set, that relies in no way on a notion of an empty set? Surely, one of these must hold or the definitions are circular.

CRGreathouse
Homework Helper
What would be a good definition for empty set, that relies in no way on a notion of a non-empty set, or a definition of a non-empty set, that relies in no way on a notion of an empty set? Surely, one of these must hold or the definitions are circular.
The empty set is the set that contains exactly those elements that are not equal to themselves.

HallsofIvy