Can the empty set be defined as a contradiction in logic?

Question

Can the empty set be defined as something like this in logic? $$ \square\left(\exists x\mid E(x)\iff \exists x: \neg x=x\right) $$ Also, how should the empty set be defined in logic? Thanks and simultaneously sorry for my amateurish question.

Answer 1

The null set can be defined using a contradiction, but not explicitly as a contradiction. That is, if we have that a set $A$ already exists, then we can use the Axiom of Separation to get the subset of $A$ that contains exactly those members of $x\in A$ such that $x\not=x$. Since nothing fits this requirement, it follows that the subset of $A$ in question is empty. Then, you can use the Axiom of Extensionality to prove its uniqueness.

Answer 2

In axiomatic set theory we typically just stipulate the existence of an empty set by something like $\exists x \forall y \ y \not \in x$. (or, provide some set of axioms from which this statement can be derived). So the formula $\forall y \ y \not \in x$ can be used as a formula ('predicate') to say that $x$ is empty. If you insist on using a predicate symbol $E$ where you want to use $E(x)$ to say that set $x$ is empty, you can do something like $$\forall x (E(x) \leftrightarrow \forall y \ y \not \in x)$$

Note that an empty set is of course not the same as a contradiction ... but I am guessing you are thinking something along the lines of: if an empty set would contain something, we would obtain a contradiction ... you could capture that idea by: $$\forall x (E(x) \leftrightarrow \forall y (y \in x \to \bot))$$

Answer 3

While a contradiction is obtained as an intermediate result in a proof of the existence of an empty set (see line 10 below), its existence should not be thought of as a contradiction in itself.

Further to other answers here, given the existence of any set, it is also possible to prove the the existence of an empty set without the use of the equality relation and reflexivity. (Saves 4 lines.) Using a form of natural deduction, we have the following text version of a proof with side scroll bar:

Let x be any set.

1   Set(x)
    Premise

Apply Subset Axiom

2   EXIST(a):[Set(a) & ALL(b):[b in a <=> b in x & ~b in x]]
    Subset, 1

3   Set(y) & ALL(b):[b in y <=> b in x & ~b in x]
    E Spec, 2

4   Set(y)
    Split, 3

5   ALL(b):[b in y <=> b in x & ~b in x]
    Split, 3

    Suppose...

    6   z in y
        Premise

    7   z in y <=> z in x & ~z in x
        U Spec, 5

    8   [z in y => z in x & ~z in x] & [z in x & ~z in x => z in y]
        Iff-And, 7

    9   z in y => z in x & ~z in x
        Split, 8

    Obtain the contradiction... 

    10  z in x & ~z in x
        Detach, 9, 6

11  ~EXIST(c):c in y
    Conclusion, 6

12  Set(y) & ~EXIST(c):c in y
    Join, 4, 11

As Required:

13  ALL(a):[Set(a) => EXIST(b):[Set(b) & ~EXIST(c):c in b]]
    Conclusion, 1

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Answer 4

not necessarily. Technically it could be the case that: $\exists$ x (Ex $\land$ x = x),[which is actually quite intuitive, it seems exaggerated to deny that the set of all prime even numbers is identical to itself, even if it is obviously empty], or that an empty set doesnt entail or is equivalent to any other contradiction. One could get a contradiction to entail an empty set, such as claiming that, given a set A in which u are an element of it if and only if u r a true contradiction, that set A will be empty, as in:

($\exists$ A(x $\in$ A $\iff$ Dx) $\to$ $\exists$ A(A = {$\emptyset$})}, in which "Dx" means "x is a Dialetheia", and Dialetheia is the term for a true contradiction. But im not sure if we could actually define an empty set as a contradiction. It generally can be used to denote contradictions, such as the set of all pieces of matter who dont occupy space or the set of all feline humans, but the notion of an empty set is not itself defined as a contradiction. Really it just means a set which has no elements, or:

$\forall$ x(Ex $\iff$ ($\forall$ y(y $\notin$ x))).

Surely all sets with contradictory properties will be empty sets, but it is not necessarily (as a first impression) the case that there is no set which lacks contradictory properties and is an empty set, which would be necessary if an empty set definitionally (and thus necessarily) has atleast one contradictory property.