lel postulate, and hyperbolic geometry is a model that incorporates the
negation of the parallel postulate.The Continuum: Where Are We Now?
One of the preeminent physicists of today, John Archibald Wheeler (whom
we shall encounter when we discuss quantum mechanics), feels that both
the discrete structure of the integers and the fundamental nature of the
continuum are vital to the work of physics, and weighs in with a physi-
cist’s point of view.
For the advancing army of physics, battling for many a decade
with heat and sound, fields and particles, gravitation and space-
time geometry, the cavalry of mathematics, galloping out ahead,
provided what it thought to be the rationale for the real number
system. Encounter with the quantum has taught us, however, that
we acquire our knowledge in bits; that the continuum is forever
beyond our reach. Yet for daily work the concept of the continuum
has been and will continue to be as indispensable for physics as it
is for mathematics. In either field of endeavor, in any given enter-
prise, we can adopt the continuum and give up absolute logical
rigor, or adopt rigor and give up the continuum, but we can’t pur-
sue both approaches at the same time in the same application.^10Wheeler sees a clash between the current quantum view of reality
(Wheeler’s absolute logical rigor) and the continuum, a useful mathe-
matical idealization that can never be. Mathematicians are lucky—they
do not have to decide whether the object of their investigation is either
useful or a great description of reality. They merely have to decide if it is
interesting.
Given Cohen’s result on the undecidability of CH within ZFC, and since
CH is independent of ZFC, what are the choices for continuing research?
The problem has basically been removed from the domain of the mathe-
matician, most of whom are content with ZFC as an axiomatic frame-
work. The majority of logicians concentrate on the ZFC part of the problem,
and much work is being done on constructing other axioms for set theory
in which CH is true. Future generations of mathematicians may well de-
cide to change the industry standard, and abandon ZFC for some other
system.
Of what value is all this? From the mathematical standpoint, even
though developments in the twentieth century have diminished the im-
portance of solving Hilbert’s first problem, the continuum is one of the
The Mea sure of All Things 25