Just as there are different geometries (Euclidean, projective, spherical,
hyperbolic—to name but a few), there are different set theories. One of
the most widely studied is the axiomatic scheme proposed by Ernst
Zermelo and Adolf Fraenkel, who came up with a system to which was
added the axiom of choice.^9 The industry-standard version of set theory is
known as ZFC—the Z and F stand for you-know-who, and the C for the
axiom of choice. Mathematicians are inordinately fond of abbreviations,
as the mathematical aesthetic dictates that conveying a lot of meaning in
very few symbols is attractive, and so CH is the abbreviation for the con-
tinuum hypothesis.
The first significant dent in Hilbert’s first problem was made in 1940 by
Kurt Gödel (of whom we shall hear much more in a later chapter), who
showed that if the axioms of ZFC were consistent, then including CH
as an additional axiom to produce a larger system of axioms, denoted
ZFCCH, did not result in any contradictions, either.
This brought the continuum hypothesis, which had been under scru-
tiny by mathematicians (who would have liked either to find a set of real
numbers with a cardinal number other than aleph-0 or c, or prove that
such a set could not exist), into the realm of mathematical logic. In the
early 1960s, Paul Cohen of Stanford University shocked the mathematical
community with two epic results. He showed that if ZFC were consistent,
CH was undecidable within that system; that is, the truth of CH could
not be determined using the logic and axioms of ZFC. Cohen also showed
that including the negation of CH (abbreviated “not CH”) to ZFC to produce
the system ZFC not CH was also consistent. In conjunction with Gödel’s
earlier result, this showed that it didn’t matter whether you assumed CH
was true or CH was false, adding it to an assumed-to-be-consistent ZFC
produced a theory that was also consistent. In the language of mathemat-
ical logic, CH was independent of ZFC. This work was deemed so sig-
nif icant that Cohen (who passed away in the spring of 20 07), was awarded
a Fields Medal in 1966.
What did this mean? One way to think of it is to hark back to another
situation in which an important hypothesis proved to be independent of a
prevailing set of axioms. When Euclidean geometry was subjected to in-
vestigation, it was realized that the parallel postulate (through each point
not on a given line l, one and only one line parallel to l can be drawn) was
independent of the other axioms. Standard plane geometry incorporates
the parallel postulate, but there exist other geometries in which the paral-
lel postulate is false—in hyperbolic geometry, there are at least two lines
that can be drawn through any point off the line l that are parallel to l.
Logicians say that plane geometry is a model that incorporates the paral-
24 How Math Explains the World