Universal set

For "universal set" in the sense of a universe of discourse with respect to which the absolute set complement is taken, see Universe (mathematics).

In set theory, a universal set is a set which contains all objects, including itself.[1] In set theory as usually formulated, the conception of a set of all sets leads to a paradox. The reason for this lies with Zermelo's axiom of comprehension: for any formula \varphi(x) and set A, the set

\{x \in A \mid \varphi(x)\}

which contains exactly those elements x of A that satisfy \varphi exists. If the universal set V existed and the axiom of separation applied to it, then Russell's paradox would arise from

\{x \in V\mid x\not\in x\}.

More generally, for any set A we can prove that

\{x \in A\mid x\not\in x\}

is not an element of A.

A second issue is that the power set of the set of all sets would be a subset of the set of all sets, providing that both exist. This conflicts with Cantor's theorem that the power set of any set (whether infinite or not) always has strictly higher cardinality than the set itself.

The idea of a universal set seems intuitively desirable in the Zermelo–Fraenkel set theory, particularly because most versions of this theory do allow the use of quantifiers over all sets (see universal quantifier). This is handled by allowing carefully circumscribed mention of V and similar large collections as proper classes. In theories in which the universe is a proper class, V \in V is not true because proper classes cannot be elements.

Contents

Set theories with a universal set

There are set theories known to be consistent (if the usual set theory is consistent) in which the universal set V does exist (and V \in V is true). In these theories, Zermelo's axiom of separation does not hold in general, and the axiom of comprehension of naive set theory is restricted in a different way.

The most widely studied set theory with a universal set is Willard Van Orman Quine’s New Foundations. Alonzo Church and Arnold Oberschelp also published work on such set theories. Church speculated that his theory might be extended in a manner consistent with Quine’s,[2] but this is not possible for Oberschelp’s, since in it the singleton function is provably a set,[3] which leads immediately to paradox in New Foundations.[4]

Zermelo–Fraenkel set theory and related set theories, which are based on the idea of the cumulative hierarchy, do not allow for the existence of a universal set.

Uses in other areas of mathematics

In probability theory and random variables, the universal set or sample space is the one that contains all possible events.

See also

References

  1. ^ Forster 1995 p. 1.
  2. ^ Church 1974 p. 308. See also Forster 1995 p. 136 or 2001 p. 17.
  3. ^ Oberschelp 1973 p. 40.
  4. ^ Holmes 1998 p. 110.

Bibliography