There was a time back at the university, when pretty much any question about mathematics I came up with led to the axiom of choice. This seemingly innocuous and at first sight quite reasonable axiom leads to pretty strange conclusions such as the Banach–Tarski paradox. Roughly speaking, it requires that for any collection of non-empty sets it should be possible to construct an element of the Cartesian product.
Here are two of the questions I had at the time:
- Can we construct a basis of the reals as a vector space over the rationals? (See Mathoverflow for the same question asked by somebody else and pretty much the same answers I got from Usenet at my time.)
- If we have a surjective map from a set A to a set B and also a surjective map from B to A, can we construct an isomorphism of A and B? (Here is the old discussion.)
Now, by their nature, axioms are our basic assumptions of what the rules of a mathematical theory should be. As such, they cannot be proven and need some faith. There is a wonderful philosophical take on this in the paper “Believing the Axioms” by Penelope Maddy.
If enough people have faith in something, they could start a religion. For example, in censuses in New Zealand and Australia there was a joking effort to make enough people state “Jediism” as their religion to make it an official religion.
So will it be possible to find enough mathematicians to make their faith in the axiom of choice their official religion to start the Church of Believers in the Axiom of Choice?