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Is there any axiom or theorem of any part of math/logic that states the fact:

"Every assumption about the elements of the empty set is true."

? If no, can you imagine why it is not true?

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This may be of help to what you are looking for: en.wikipedia.org/wiki/Vacuously_true –  Apostolos Jul 11 '11 at 17:29
    
Perhaps this question is somewhat related math.stackexchange.com/questions/50492/… –  Martin Sleziak Jul 11 '11 at 17:29
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My answer here might be somewhat useful to you. –  Asaf Karagila Jul 11 '11 at 18:06
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3 Answers

up vote 8 down vote accepted

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.)

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thank you, Mr. Yuan. –  RinKaMan Jul 11 '11 at 17:39
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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.)

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@RinKaMan: …and $x\in\emptyset$ is false by definition of $\emptyset$. –  beroal Jul 26 '11 at 21:18
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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.

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If you believe that statements are logically equivalent to their contrapositives, then the statement that "the current King of America has ruled for over 2000 years" is equivalent to the statement that "nobody who has not ruled for over 2000 years is the current King of America" which is surely true. –  Qiaochu Yuan Jul 11 '11 at 17:50
    
Also, if you believe that a set with $n$ elements has $2^n$ subsets, then you believe that the empty set is a subset of every set, so you believe that the set "current kings of America" is a subset of every set (such as "people who have ruled America for over 2000 years" etc.). –  Qiaochu Yuan Jul 11 '11 at 17:52
    
@Qiaochu Yuan: the issue Dedalus alludes to is the position that a "statement" can't be equivalent to its contrapositive if the statement is not even well posed. This is more of a philosophical issue because in mathematics every statement is usually taken to be true or false. But "0/0 = 1" is arguably neither true nor false, as it doesn't even make a claim if 0/0 is not defined. Similarly, "The King of America is bald" makes no claim if there is no King of America. The issue of ill-posed questions is a sort of blind spot in mathematics; we work around so well in practice without noticing them. –  Carl Mummert Jul 11 '11 at 18:23
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... I would compare things like "0/0 = 1" with absurdities like "the Jacobson radical of the color blue has positive area". –  Carl Mummert Jul 11 '11 at 18:26
    
Qiaochu: Yes, what I meant (and I think that I expressed) is just what Carl clarified there. –  Dedalus Jul 11 '11 at 18:38
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