The Continuum Hypothesis and the Axiom of Choice are two propositions that come up in the study of the foundations of mathematics. Despite one being called an axiom and the other a hypothesis they are viewed in a similar way by the mathematics community. This article is going to assume some basic knowledge of mathematics and will be awkward in places given the lack of available symbolism.
The Continuum Hypothesis (CH) states that there is no ordinal number greater than the of the set of natural numbers (N) but smaller than the real numbers (R). The natural numbers are the counting numbers; 0,1,2,3… and the real numbers are essentially all of them; pi, 5, 7/56, 39.7712… Any number you can write out as a decimal, infinitely long or otherwise is a real number. Georg Cantor proved that there are as many natural numbers as there are fractions, and he was also the first person to state the Continuum Hypothesis. The name comes from the fact that another name for the reals is “the continuum”
The hypothesis has not been proven. Cantor proved that the reals were a strictly bigger set than the naturals. Given the standard axiomatic foundations for arithmetic (known as the Zermelo-Fraenkel axioms), it turns out that CH is independent of them. That means that it can be shown that CH can be neither proven nor disproven within this system.
The Axiom of Choice (AoC) is another statement which turns out to be independent of the Zermelo-Fraenkel axioms. Roughly put, it states that given a family of (disjoint) sets, there is a way to pick exactly one element from each set. Imagine the case of shoes. Even if you had an infinite number of shoes you could always pick the left shoe. There is a rule to pick out exactly one element of each set (a set here meaning a pair of shoes). Now, in the case of socks things are a little trickier. You don’t get “left” and “right” socks, so the method for picking out exactly one element is not at all obvious. This should give an indication of where this problem becomes a little tricky.
Many many equivalent definitions of the axiom of choice have been given. The one with the most fun name is “Zorn’s Lemma.” This leads to the popular mathematicians joke “What’s yellow, tangy and equivalent to the axiom of choice? Zorn’s Lemon!” Mathematicians are not without their sense of humour…
It turns out that the continuum hypothesis is independent of ZFC, which is the ZF axioms with AoC. However, ZF axioms with CH are sufficient to prove AoC. (Strictly speaking it’s the generalised Continuum Hypothesis, but we’ll ignore the distinction for now).
The AoC is generally accepted to be true by mathematicians because it is very useful in proving otherwise unattainable results in interesting areas of mathematics. It is unpopular among some mathematicians, however, because it implies some bizarre things, like the Banach-Tarski paradox which says that you can split a sphere of radius r into several pieces and then put the pieces together so that they form two spheres of radius r. These counter intuitive constructions are only possible by making an infinitely “wiggly” line cut the sphere, so they don’t hold any secrets on how to double your money…
Both the Axiom of Choice and the Continuum Hypothesis are important and various combinations of their inclusion or negation have been explored as ways to extend arithmetic and have lead to interesting new mathematics.
