Continuum Hypothesis

The Continuum Hypothesis

Infinity has ... infinite ways to trouble our finite minds. This was proved by Georg Cantor in 1874.

The "smallest level" of infinity has to do with countable things that can be put in some order.

The natural numbers 1,2,3,4... are an example. The squares, 1,4,9,16... although a subset of the natural numbers are also in the "same level" of infinity, for there is a one-to-one correspondence between the two sets: 1 <-> 1, 2 <-> 4, 3 <-> 9, 4 <-> 16... and so a natural order (the 1st square, the 2nd etc). The two sets are considered to have the same cardinality (i.e. number of elements) called aleph₀. This seems strange: one set is a proper subset of another and still they have the same number of elements. This is exactly the definition of infinite sets.

What about rational numbers? These are a superset of the natural numbers but still of class aleph₀. It turns out that there is a way to put rational numbers in order: 1, 2, 1/2, 1/3, 3, 4, 3/2, 2/3, 1/4 ...
(the pattern is based on a diagram so it is not obvious as shown here).

Things change when we examine the real numbers. There is no way to create a complete list of reals and this was shown by Cantor with a beautiful argument, the "diagonal" one: Suppose we had such a complete list of real numbers between 0 and 1 :

r1=0.a₁₁a₁₂a₁₃...
r2=0.a₂₁a₂₂a₂₃...
r3=0.a₃₁a₃₂a₃₃...
........

Where a_ij take values in 0,1,...,9 and all numbers are written with infinite number of digits (e.g. 0.2=0.20000..., 1/7=0.142857142857...).

Then we can create another real r'=0.b₁b₂b₃.. not included in the list. Suffices to take as b₁ any number not equal to a₁₁, as b₂ any number not equal to a₂₂ etc. Therefore, the list is not complete, we have a contradiction and our assumption that there is such a list does not hold. This conclusion is easily extended to the set of all real numbers.

The above mean that the cardinality of the set of the real numbers is greater than aleph₀. Let's call it c from "continuum" to emphasize the fact that the set of real numbers is mapped onto something continuous in space, namely the line. Other sets such as the number of points in a plane, in a 3-dimensional space and in fact in a n-dimensional space also have the "continuum" cardinality. But there are many more levels of infinity (in fact infinite ones) larger than c. Take the set of all subsets of a set with cardinality c. This leads to a "larger" infinity. Repeat this procedure ad infinitum.

The interesting problem is at the lower end. Let's call aleph₁ the next (unknown) level of infinity after aleph₀. The Continuum Hypothesis states simply that aleph₁= c, that is, there is no level of infinity between the one of countable things and the one of continuous ones. Why is this conjecture so interesting?

First, because there does not seem to be any particular reason not to have some unknown class of infinity in between - remember, we are talking about infinity, intuition is usually wrong.
Second, because moving from an infinity of points (aleph₀) to the continuum (c) is a giant step indeed - there is a feeling that something is missing.
Third, the question has to do with our ability to understand infinity and its various forms, and possibly discover others. An infinite collection of points is a "quantised" world. The line is a "fractional" world (zoom-in as you wish, you get the same picture). Could there be something in the middle? Where would you put the nature of time for example?

So this was the first of the 23 problems - milestones that David Hilbert suggested in 1900, at the 2nd International Conference on Mathematics in Paris, that they should define research in mathematics for the new century (and indeed, it is not an exaggeration to say that modern mathematics largely come from the attempts to solve these 23 problems).

Where are we now:

Kurt Gödel proved in 1940 that the Continuum Hypothesis (CH) is consistent with the accepted axioms of set theory (the Zermelo-Fraenkel-Skolem system). But in 1963 Paul Cohen proved that the reverse also holds, that is the refutation of the CH is also consistent with the same axioms. This means that the Hypothesis is undecidable within set theory.

In essence, the situation is analogous with the famous 5th (parallels) Euclidean axiom. In principle, one can extend the axioms of set theory accepting or rejecting the Continuum Hypothesis and obtain different theories, in the same way we have Euclidean and non-Euclidean geometries. However, mathematicians are reluctant to do so. Set theory is so fundamental to mathematics, we cannot afford different "versions".

On the other hand, there is simply no evidence on which way would be more "reasonable". The search is for an equivalent form which would at least be more intuitively appealing to add as an axiom. Of course there is also much debate on whether there is any need at all to extend the axioms of set theory.