The Continuum Hypothesis, explained
It might sound like an innocuous question: How many real numbers are there? Georg Cantor’s inquiry about the size of the continuum sparked an amazing development of technologies in modern set theory, and influences the philosophical debate until this very day.
The continuum hypothesis
What is the continuum hypothesis? Very roughly speaking, the continuum hypothesis is a statement about the behaviour of certain infinite numbers — the so-called cardinals. The finite cardinals are very familiar: 0, 1, 2, 3, ... They provide answers to “how many?”-questions, such as “How many solutions does this equation have?” Or, “how many elements does this set have?”
It turns out that the finite cardinals are not enough: An equation might have infinitely many solutions, a set might contain infinitely many numbers. At first view, it may seem as if the answer to a “how many?”-question in such cases is just “infinitely many.” Cantor, however, showed in 1879 that there is not just one infinity: He proved that there are more real numbers than there are natural numbers.
Both the set of real numbers and the set of natural numbers are infinite. So how can we say that there are more reals than there are naturals? Cantor showed that it is impossible to assign exactly one real number to each natural number in such a way that every real number gets assigned. In other words, each such assignment will leave many real numbers unassigned. (Mathematically speaking, there’s no injection from the reals to the naturals.)
This observation has big implications: There must be at least two different infinities, one bigger than the other. And, it turns out that there are many more infinite numbers than there are finite numbers: the so-called aleph numbers, denoted by the Hebrew letter א (“aleph”).
The smallest infinite cardinality is א0 (“aleph 0”). That’s exactly the size of the set of natural numbers. The next bigger cardinal number is א1 (“aleph 1”), then א2, א3, and so on. What’s the cardinality of the set of real numbers? Cantor formulated one possible answer in his famous continuum hypothesis. This is one way to state it:
Every infinite set of real numbers is either of the size of the natural numbers or of the size of the real numbers.
The continuum hypothesis is, in fact, equivalent to saying that the real numbers have cardinality א1. If the continuum hypothesis is false, it means that there is a set of real numbers that is bigger than the set of natural numbers but smaller than the set of real numbers. In this case, the cardinality of the set of real numbers must be at least א2.
For many years, mathematicians tried to determine whether the continuum hypothesis is true or false. The problem was so pressing that David Hilbert made it the first on his list of 23 problems, published in 1900. But it took until the 1930s before significant progress could be made.
Kurt Gödel proved in 1938 that the continuum hypothesis is consistent with the ZFC-axioms of set theory — those axioms on which mathematicians can base their everyday reasoning. Gödel showed that adding the continuum hypothesis to these axioms does not result in a contradiction. This is far from proving that the continuum hypothesis is true. Such a proof would describe how the truth of the continuum hypothesis follows from the axioms of set theory.
Gödel proved his consistency result by constructing a set-theoretic world in which the continuum hypothesis is true: the so-called constructible universe. From the existence of this universe, we can learn a bit more about the status of the continuum hypothesis. Imagine there was a proof, from the axioms of set theory, that the continuum hypothesis is false. As the axioms of set theory hold in Gödel’s constructible universe, it must follow that the continuum hypothesis is false in this universe. But Gödel showed that it’s true — a contradiction!
Gödel’s consistency result therefore implies that there cannot be a proof that the continuum hypothesis is false. But can we find a proof that it is true?
It took almost another 30 years to answer that question. Paul Cohen proved in the 1960s that it is consistent with the axioms of set theory that the continuum hypothesis is false — a result for which he received the Fields Medal in 1966, one of the highest honours for a mathematician.
To prove this result, Cohen invented a new method of constructing set-theoretic universes: the forcing technique. Using this technique, he constructed a set-theoretic universe in which the continuum hypothesis is false. As in Gödel’s case, this result implies that there can be no proof, from the axioms of set theory, that the continuum hypothesis is true.
Where does this leave us? Combining these results of Gödel and Cohen, we know that there can neither be a proof, from the axioms of set theory, showing that the continuum hypothesis is true nor that the continuum hypothesis is false. That is to say, the continuum hypothesis is independent of the axioms of set theory: these axioms are not strong enough to determine whether the continuum hypothesis is true or false.
Can we settle the continuum hypothesis?
Up to this very day, the results of Gödel and Cohen influence contemporary set theory and the philosophical debates surrounding it. Every day, set theorists are building new set-theoretic universes by using, essentially, the techniques that Gödel and Cohen developed. But that’s only the mathematical side of it.
A crucial philosophical debate is still running: What’s the answer to Cantor’s continuum question? Is the continuum hypothesis true? Or is it false? After all, the results of Gödel and Cohen only show that it is impossible to find a proof from the ZFC-axioms of set theory. So maybe an answer is still out there?
Some set theorists believe that they can find new axioms for set theory that will allow mathematicians to settle the continuum hypothesis. These set theorists are often called universists because they believe, roughly, that there is just one true mathematical universe. Just like physicists aim to describe the properties of the physical universe surrounding us, the universists believe that there is one mathematical reality that they are trying to describe.
Under this universe view, every mathematical statement is either true or false. So, to learn whether or not the continuum hypothesis is true, we just have to find out more about this mathematical reality, the mathematical universe.
But how can we find out more? By finding new axioms that describe the true universe better, in a more complete way, than the already established ZFC-axioms of set theory do. Universists hope that such axioms will eventually allow for a proof of either the continuum hypothesis or its negation.
Not all set theorists are convinced by the universe view just described. Joel Hamkins, professor of logic at Oxford university, believes that there are many equally important set-theoretic universes. The techniques developed by Gödel and Cohen, and further developed by many others around the world, allowed set theorists to peek into vastly different mathematical universes. According to Hamkins, all these universes together form the set-theoretic multiverse.
On this multiverse view, there is not one but very many true set-theoretic universes. In some of these universes, the continuum hypothesis is true. In others, it’s false. Hamkins therefore argues that the continuum question is answered:
“I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for [by the universists].” — Joel Hamkins (The set-theoretic multiverse)
Set theorists have accumulated extensive knowledge about how the continuum hypothesis behaves in many different universes of set theory. They understand precisely when it is true, or false, and how big or small the continuum can possibly be. In Hamkins’s view, this knowledges forms the answer to the continuum question — it’s not a simple “yes” or “no.”
What do you think? Should we be able to find out whether the continuum hypothesis is true or false? Or do we already know everything there is to know? The continuum hypothesis is still at the centre of one of the most exciting debates in contemporary set theory.
