Assumption about elements of the empty set

Solution 1:

The correct statement is that every universal statement about the elements of the empty set is true; this is known as vacuous truth. (One might say that universal statements are "true until proven false." Alternately, the negation of every universal statement is an existential statement, and they should all be false for the empty set, so every universal statement should be true for the empty set.)

Solution 2:

If you mean statements of the form $x\in\emptyset \Rightarrow P(x)$, then they are always true. (Since implication is true whenever antecedent is false.)

Solution 3:

I have no idea whether any such theorem exists to be honest, but here is my thinking regarding it.

Let's try to switch from a moment, from talking about elements of the empty set, to talking about the famous current king of America (no, not the president, the king). You've never heard of him? Oh well, let me tell you somethings about him. He have ruled America for over 2000 years, and he's the wisest person there is, and he's around 2 meters tall, with dark hair. Now, are these true statements? I mean, the king of America doesn't exist, so what truth value can we ascribe to these statements? This relates to definite description, a concept in logic. Bertrand Russell wondered whether such statements are true, false or simply without meaning. My personal opinion on the matter is that these statements don't have a truth value, they're just meaningless statements.

Now, for elements of the empty set, I think something similar applies. If you say that every element of the empty set has property X, then, I believe it is meaningless.