We have seen in the Fun Fact Cantor Diagonalization
that the real numbers (the "continuum") cannot be placed in
11 correspondence with the rational numbers. So
they form an infinite set of a different "size"
than the rationals, which are countable. It is not
hard to show that the set of all subsets
(called the power set) of the rationals has
the same "size" as the reals.
But is there a "size" of infinity between
the rationals and the reals? Cantor conjectured that
the answer is no. This came to be known as the
Continuum Hypothesis.
Many people tried to answer this question in the early
part of this century. But the question turns out to
be PROVABLY undecidable! In other words, the
statement is indepedent of the usual axioms of set theory!
It is possible to prove that adding the
Continuum Hypothesis or its negation would not cause
a contradiction.
So, you can take either the Continuum Hypothesis
or its negation to be true, and it would not affect
the truth of other statements in mathematics!
Presentation Suggestions:
Students will find it amazing that statements that
seem to have an answer may in fact be taken to be either
true or false, depending on the model of the real numbers
that you use!
The Math Behind the Fact:
This is deep set theory.
K. Godel and later, P. Cohen showed the independence of
the Continuum Hypothesis from the ZermeloFraenkel Axioms
of Set Theory.
The "size" of a set is called its cardinality.
How to Cite this Page:
Su, Francis E., et al. "Continuum Hypothesis."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
