Why must it be either true or false that the empty set is a subset of all sets?

Question

I encountered a proof that the empty set is a subset of every set via this comment(Is "The empty set is a subset of any set" a convention?) which shows that it cannot be false that the empty set is a subset of every set. Without necessarily going into a proof of how the empty set is a subset of every set, I was wondering why the fact that it cannot be false that the empty set is a subset of every set shows that this is true- could it not be the case that the concept of subsets is meaningless with regards to the empty set, and it is not enough to show that it could not be false; that this statement could neither true or false as it has no meaning in this context? Also, I would appreciate some explanation as to how this condition holds "vacuously" as far as terminology, as I have learned that for an implication to be vacuously true, it is true when it's hypothesis is false.

Thanks

Answer 1

You can prove that the empty set is a subset of every set by going to the definition. In one sense that says it is not a convention, but the edge cases of definitions are chosen to make things as clean as possible. One could claim that it is a convention that subset is defined this way. In this case it seems very natural to have the definition so that it is true.

The definition of $A \subset B$ is that all elements of $A$ are also elements of $B$. For the empty set we say it is vacuously true because there are no elements of $\emptyset$. If we expand the abbreviation of subset we get $\forall x (x\in A \implies x \in B)$ As there are no $x \in \emptyset$ when $A=\emptyset$ the antecedent is always false so the implication is always true. That is what we mean by vacuous truth.

Answer 2

I think invoking the law of the excluded middle is unnecessary here; see the accepted answer of Is "The empty set is a subset of any set" a convention? or Ross Millikan's answer here.

Note that the proof I have just recommended is also the first proof that Halmos gives, and he says it is a correct proof. He also says it is "perhaps unsatisfying"; why? I suppose this is because he expects some readers not to have become familiar with vacuously true statements prior to reading this passage.

But it's unclear to me why he says we should prove that $\emptyset \subset A$ cannot be false. I would instead prove that $\lnot(\emptyset \subset A)$ is false, because in the general case of proving some assemblage of words and symbols I agree with you that the thing to be "proved" might not be a mathematical statement after all, hence neither true nor false mathematically. But if we fully accept the law of the excluded middle, then by disproving $\lnot P$ we prove $P$.

Answer 3

Why must it be either true or false that the empty set is a subset of all sets?

Your questions is equivalent to "Why do we accept the empty set everywhere any non-empty set is allowed?" or "Why do we consider the empty set to be a set?".

Practically, it is this way because excluding the empty set from set-theoretical considerations would make any set theory incredibly complicated, while solving no problem.

Any and all definitions and operations we have would become more complicated if we had no $\emptyset$: while we would not mention it per se (as we would not have the notation or the concept), we still would to, for example, need to talk about the case of two sets not intersecting. Instead of saying $A \cap B = \emptyset$ we would need to write "$A$ and $B$ do not intersect" or "$A$ and $B$ have no element in common" or $\not \exists x: x \in A \land x \in B$ and so on and forth.

Much worse, in any composite formula where a sub-expression could evaluate to the empty set, we would need step out of the world of mathematical formulation, and would have to speak in human language about it (or in explicit $\exists, \forall$ notation). Even just a way to formulate $(A \cap B) \subset C$ in a way that never lets the case $A \cap B = \emptyset$ occur would be complicated, or at least laborious.

In short, it would be a nightmare - for no gain whatsoever. At no point does $\emptyset$ lead to any contradictions or complications; all definitions and axioms can be and easily are formulated in a way that our conventional usage of $\emptyset$ just follows automatically. Not allowing $\emptyset$ would be like arbitrarily introducing "division-by-zero-style" exceptions to all set theoretical statements.

It is important enough that the usual set theory has an Axiom of the Empty Set.

It is equivalent of allowing addition and subtraction of 0, or multiplication and division by 1. On the face of it doing so may be nonsensical, but if you do not consider it as part of the mathematical structures you very quickly get into deep trouble, indeed.

Formally, the empty set is still a set. It has properties (like $|\emptyset|=0$). In mathematical terms, it can be seen as part of the category of sets and plays an important role there; in Computer Science terms it would be an object of type "set" and plays an important part in all "typed" systems.

Philosophically, the empty set is not nothing. It is like an empty bag or an empty bank account: such a thing certainly exists and we have no issues imagining or working with it, it just does not contain anything.

---

As to your other question regarding "vacuous truth": statements like $\forall Set(X): \emptyset \subset X$ is in itself not vacuous; it is simply true on this level of formulation.

"Vacuous" means that any statement of the form $\forall x \in \emptyset : P(x)$ is true (by convention or definition). It is relatively natural to set this value to true if you look at the representation $\neg (\exists x \in \emptyset : \neg P(x))$ which is equivalent to the previous statement. Intuitively, $\exists x \in \emptyset$ is immediately "false" (it is not a statement, so it is "false in quotes"); so to assign the true value to the opposite (our original vacuous statement) makes the most sense.

Note that "vacuous truth" is not something weird or illogical, it just is a label for something which is true but gives no additional meaning or semantic power - it's useless, but harmless; but not having (the concept of) it would be harmful.