Today we’ve got a bit of a treat. I’ve been holding off on this for a while, because I wanted to do it justice. This isn’t the typical wankish crackpottery, but rather a deep and interesting bit of crackpottery. A reader sent me a link to a website of a mathematics professor, N. J. Wildberger, at the University of New South Wales, which contains a long, elegant screed against the evils of set theory, titled “Set Theory: Should You Believe?”

It’s an interesting article – and I don’t mean that sarcastically. It’s over the top, to the point of extreme silliness in places, but the basic idea of it is not entirely unreasonable. Back at the beginnings of set theory, when Cantor was first establishing his ideas, there was a lot of opposition to it. In
my opinion, the most interesting and credible critics of set theory were the constructivists. Put briefly, constructivists believe that all valid math is based on constructing things. If something exists, you
can show a concrete instance of it. If you can describe it, but you can’t build it, then
it’s just an artifact of logic.

Some of that opposition continues to this day, and it’s not just the domain of nuts. There are
serious mathematicians who’ve questioned the meaningfulness of some of the artifacts of modern
set-theory based mathematics. Just to give one prominent example, Greg Chaitin has given lectures in which he discusses the idea that the real numbers aren’t real: they’re just logical artifacts which can never actually occur in the real world, and rationals are the only real real numbers. (I don’t think that Greg really believes that – just that he thinks it’s an interesting idea to consider. He’s far
too entranced with set theory. But he clearly considers it valid enough to be worth thinking about
and talking about.)

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