# The Universal Set

Posted in: Math , Philosophy

In von Neumann set theory the universe $$V$$, which is the class of all sets, exists. the complement $$\sim A$$ of a set $$A$$ can then be defined as

$$\sim A = V \sim A$$.

But this is not possible in Zermelo-Fraenkel set theory, and it may be of interest to see why not in some detail. (…) [For this] we would need to prove:

(1) $$(\exists ! B)(\forall x)(x \in B \leftrightarrow x \notin A)$$

and then we would define complementation by:

(2) $$\sim A = y \leftrightarrow (\forall x)(x \in y \leftrightarrow x \notin A)\ \&\ y\text{ is a set}$$.

Suppose now that it were possible to prove (1). Let $$A = \emptyset$$, then:

(3) $$(\exists ! B)(\forall x)(x \in B)$$

That is, $$B$$ is the universal set to which every object belongs; but with $$B$$ at hand the axiom schema of separation reduces to the axiom schema of abstraction by taking $$A$$ as the universal set $$B$$, and Russell’s paradox may be derived. We conclude that (1) cannot be proved and Definition (2) is impossible in Zermelo-Fraenkel set theory. One aspect of this discussion may be formalized in the useful result that there does not exist a universal set. As just indicated, the proof of this theorem proceeds by the line of argument of Russell’s paradox via a reductio ad absurdum.

Theorem 41: $$(\nexists A)(\forall x)(x \in A)$$

(…)

Theorem 49: $$\emptyset = \{x : x \not= x\}$$

Proof:

Suppose there were a $$y$$ such that

$$y \in \{x : x \not= x\}$$

then, by theorem 47

$$y \not= y$$

which is absurd.

Similar to Theorem 41, we also have:

Theorem 50: $$\emptyset = \{x : x = x\}$$

Taken from Axiomatic Set Theory by Patrick Suppes.