Let us define: $\bigcap \emptyset = \{ x|\forall A(A \in \emptyset \Rightarrow x \in A)\} $
I understood that this cannot be defined. Somehow it enables Russell's paradox to exist.
Why is that?
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2This might be related. – EuYu Feb 02 '14 at 13:09
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1@abiessu What you described is called vacuous truth. That implies the statement is true for whatever $x$ you choose, which in turn implies every $x$ is in the set. That's what poses the problem: there is no set containing everything. – EuYu Feb 02 '14 at 13:12
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@EuYu: I see that now, thank you for the link to the very well-explained reasoning. – abiessu Feb 02 '14 at 13:14
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You can define that in some set theories. See for example Mark Holmes' Elementary Set Theory with a Universal Set. The choice to operate in ZF set theory, which does not have such a set, is not mandatory. – MJD Mar 12 '14 at 21:29
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There is more than one theory that attempts to describe sets, and some of them are capable of expressing the set you describe, while some are not.
There are at least two reasons why the intersection of the empty set should be the set of all sets:
- In $\{x : \forall A[A \in \emptyset \implies x \in A]\}$, the precondition of the implication is always false so the implication is always true, so all $x$ satisfy the condition,
- We regard it as an essential defining property of $\bigcap$ that $\bigcap (x \cup y) = \bigcap x \cap \bigcap y$. For this to hold even when $y$ is empty, we need $\bigcap y$ to be the set of all sets.
In the most common set theory in use, we also have the Axiom of Separation (or, pedantically, the axiom schema of separation), that states for any set $x$ and property $P$ that you specify, there is a set of "elements of $x$ that satisfy $P$". Now, if you have $x$ permitted to be a set containing all sets, this reduces to "for any $P$, there is a set containing all elements with property $P$", and taking $P$ to be $y \not\in y$ we obtain Russell's paradox.
Therefore, if you have such an axiom that allows unrestricted subsets, you cannot permit the set of all sets (and hence, the intersection of the empty set) without encountering Russell's paradox.
In other set theories, the set of all sets is permissible, but there must be restrictions on what subsets can be taken in order to prevent Russell's paradox. For example, we may reject the formula $x\not\in x$ syntactically as giving conflicting roles for $x$ as both an element and a set (...clearly sets can be elements, but one might decide that the universe ought to be separated into elements, sets-of-elements, sets-of-sets-of-elements, and so on, and clearly $x$ can have no single designation in this sense).
Ben Millwood
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