ZFC

Jerry E. Vaughan

University of North Carolina at Greensboro,
Greensboro, NC 27412, USA

Zermelo-Fraenkel set theory (denoted ZF) is a first order formal system in mathematical logic. The formal system ZF with the addition of the axiom of choice (denoted ZFC) is generally held to encompass essentially all of mathematics.

In topology (and elsewhere) there are simple statements that can neither be proved nor disproved (in ZFC). Probably the best known statement of this kind in mathematics is Cantor's continuum hypothesis:

(CH) Every infinite subset of the real numbers is either countable, or the same cardinality as the set of all real numbers.

The following is a statement in topology that can neither be proved nor disproved:

(O-W) Every countably compact, perfectly normal space is compact.

A. Ostaszewskii proved that the statement (O-W) is false assuming the combinatorial statement "Diamond", and W. Weiss proved (O-W) is true assuming that the small cardinal p is greater than the first uncountable cardinal (this is known to follow from "Martin's Axion and the negation of the continuum hypothesis").

For more about mathematical logic, we recommend the book An Introduction to Mathematical Logic, by Richard E. Hodel, PWS Publishing Co., Boston, 1995. A discussion on ZFC is given in Section 6.4.

Received by the editors: December 19, 1995.