1

Let A be a set. Because the empty set has no elements it is unfeasible to find any element that belongs to the empty set but does not belong to set A, therefore ∅⊊A is false and the opposite must be true that ∅⊆A.

However, going the other way around yields a paradox. Because the empty set has no elements it is unfeasible to find any element (let alone "all") that belongs to the empty set yet also belongs to set A, therefore ∅⊆A is false and the opposite must be true that ∅⊊A.

It cannot be that ∅⊆A is true yet also false, that is a paradox.

So we possess a paradox yet we say that it is true that the empty set is a subset of every set. Where is my logic breaking apart? What am I missing?

RandomUser
  • 143
  • 5
    You seem to be assuming that for a statement to be true for "all elements", it must also be true for "some element". But that's not always the case--there might not be any elements at all. – Eric Wofsey Feb 10 '22 at 00:24
  • 2
    You may find the answers at https://math.stackexchange.com/questions/1953218/is-the-empty-set-is-a-subset-of-any-set-a-convention and https://math.stackexchange.com/questions/2373457/confusion-in-understanding-why-empty-set-is-a-subset-of-every-set?noredirect=1&lq=1 helpful. – Eric Wofsey Feb 10 '22 at 00:24
  • I don't know guys, I'm not getting it – RandomUser Feb 10 '22 at 00:33
  • If $A \subset \emptyset$ then $A$ is also empty. You are finding a contradiction because you're assuming that a subset of the empty set can have an element. That's not true. – CyclotomicField Feb 10 '22 at 01:01

2 Answers2

3

You're missing the concept of vacuous truth; that any predicate is considered true for “all” members of an empty set.

Dan
  • 18,262
3

Because the empty set has no elements it is unfeasible to find any element (let alone "all") that belongs to the empty set yet also belongs to set A,

The definition of all does not require there to be any.   If there are none, then none are all there be.

We say all of the elements in $\varnothing$ are in $A$, because it is impossible to find an element of $\varnothing$ which is not in $A$.

$$\forall x~(x\in\varnothing \to x\in A)\iff \lnot\exists x~(x\in\varnothing\land x\notin A)$$

Graham Kemp
  • 133,231
  • 1
    I feel like it's a little weird to say that "all" can actually mean "nothing", as to me "all" at the very least implies that there is something. Is there a formal definition of "all" in set theory or is this just something that is conventionally agreed upon? – RandomUser Feb 10 '22 at 13:09