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I was wondering if one could assign random attributes to the elements empty sets, even contradictions. Because there are no elements, I can say something like:

$\forall x \in \emptyset.P(x) \land \neg P(x)$

Hence a set containing contradictions for all elements must be empty, I can assign any attribute (even its opposite).

Is my reasoning wrong, or is this just the way it is and it shouldn't be mind-boggling?

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I don't have a background in logic (at all), but I'm pretty sure that for all $x\in\varnothing$, it is true that $Q(x)$ regardless of the details of $Q$. We certainly act like this is true in less formal contexts. –  Eric Stucky Dec 27 '12 at 14:45

2 Answers 2

up vote 6 down vote accepted

"Hence a set containing contradictions for all elements must be empty"

The empty set doesn't contain any element, so it doesn't contain propositions that are contradictions.

But it is true that $\forall x \in \varnothing : P(x) \land Q(x)$. That is, $$\forall x: x\in \varnothing \rightarrow (P(x) \land \lnot P(x)).$$ Since there is no $x \in \varnothing$, $x\in \varnothing \rightarrow (P(x) \land \lnot P(x))$ is vacuously true for any (every) such (nonexisting) $x$.

Perhaps what you mean to be saying is:

  • $\varnothing = \{x \mid P(x) \land \neg P(x)\}:\;\;$ "The set of all $x$ such that $P(x) \land \lnot P(x)$."

    Since there is no $x$ such that $P(x) \land \lnot P(x)$, the empty set remains empty.

The empty set can be defined by any condition(s) that no element can satisfy.

Examples:

  • $\varnothing = \{x\in \mathbb{R} \mid x^2 = -1\}$

  • $\varnothing = \{x \mid x + 3 = x\}$

  • See this post, too.


Note: 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." Since there are NO elements in the empty set, such statements cannot be proven false (as there is no element to serve as a counterexample).

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Would the following statement then be true, or is it not even a proper statement? All elements of the empty set are even numbers. –  xfoo Dec 27 '12 at 14:51
    
Yes, that would be vacuously true: as would be "all elements of the empty set are odd numbers." –  amWhy Dec 27 '12 at 14:53
    
Thank you very much, I didn't came across the concept of vacuous truth, great explanation! –  xfoo Dec 27 '12 at 15:15
    
Christian You're welcome! Glad to help. –  amWhy Dec 27 '12 at 15:30
    
I think your first sentence is misleading. The issue is not whether the empty set contains any propositions, the issue is whether its elements can have properties that contradict each other. –  Michael Greinecker Dec 27 '12 at 15:33

It is worth pointing out that in most formalization of logic, "$\forall x\in\varnothing:P(x)$" is considered an abbreviation of $$ \forall x: (x\in\varnothing \to P(x)) $$

Since $x\in\varnothing$ is always false for every $x$, the implication $x\in\varnothing\to P(x)$ is always true (because the only way $Q\to P$ can be false is when $Q$ is true and $P$ is false), and therefore the entire quantified formula is true.

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