Continuum Hypothesis

We have seen in the Fun Fact Cantor Diagonalization that the real numbers (the “continuum”) cannot be placed in 1-1 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 Zermelo-Fraenkel 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. <https://www.math.hmc.edu/funfacts>.

References:
Any modern text on logic.

Fun Fact suggested by:
Brad Mann

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