Does the empty set exist? [closed]

Question

Does the empty set exist?

I don't think it exists

because

If there are no constituent elements of an object, it can be said that the object does not exist.

Can't this be the reason why the empty set does not exist?

Answer 1

The existence of the empty set is axiom-dependent. In typical axioms (ZF, ETCS) the empty set exists. However, there are consistent logics without an empty set. For example, take Leinster's presentation of ETCS (Leinster 2012) and remove axiom 3. Then, there is no way to construct the empty set from coproducts without appealing to higher category theory; the axioms themselves are not sufficient.

Answer 2

The empty sets existence ( in ZF) is secured by two axioms. Whether it exists as some kind of platonic object is another concern. The empty set appears in a diverse set of contexts, and so there is at least argument for its utility.

I'll make the ZF argument, so you can at least see what most people mean when they say the empty set exists.

Axiom 1: Set Existence - " There exists a set"

∃x(x=x) 

Axiom 2. Subset Axiom Scheme - "Subclasses of Sets, are Sets"

For y not free in φ i.e. φ doesn't say anything about y

∀x∃y∀z(z∈y ↔ φ(z) and z∈x) 

Putting these two together + basic logic

We get:

By Axiom 1 pick x s.t. x = x

Let φ(z) be the formula z ≠ z

Apply Axiom 2

∃y∀z(z∈y ↔ φ(z) and z∈x)

∃y∀z(z∈y ↔ z ≠ z and z∈x)

Specialize y

∀z(z∈y ↔ z ≠ z and z∈x)

As a basic axiom of equality everything is equal to itself. - More formally, we could include another axiom- extentionality which ensures all sets are equal to themselves.

So, by our Axioms

y = {z∈x | z ≠ z } exists and by the definition of equality, there are no such z's.

So, we have an empty set.

Answer 3

The word 'exist' has many meanings, so whether the empty set exists depends on which meaning of the word exist you have in mind. In mathematics, the word 'exist' can be used in a phrase such as ' three solutions exist' to refer to abstract objects that are consistent with the applicable axioms and rules, so in that sense of the word the empty set exists.

Answer 4

If no set of elements was empty, then whatever the types of whichever elements there were, every type would have at least one token. For example, if the set of unicorns isn't empty, then there is at least one unicorn; if the set of deities isn't empty, then there is at least one deity; and so on and on. Now, the empty set, of set-theoretic lore per se nota, is a set that is empty of other sets, whereas for 1 = {0} = {{}}, 1 is a set with one set in it ("the" empty one). So if the empty set didn't exist, then 1 would have a set as an element that wasn't empty as such, either, and then that set wouldn't be empty, and so on and on, so that 1 would turn out in von Neumann format to have arbitrarily many elements and subsets, or in Zermelo format just arbitrarily many subsets, neither conclusion being especially appealing, here (it seems "morally false," in the mathematical sense of that phrase, to claim that 1, as an ordinal set, is infinite, although we might imagine that it has infinitely many possible parts as a number, e.g. 1/2, 1/3, 1/4, etc.).