In 1874 the mathematician Georg Cantor published a paper that claimed to prove the existence of an infinite hierarchy of infinities, each more vast than the infinity before it, stretching out forever like some vast alien landscape. Cantor had achieved the seemingly impossible feat of counting beyond infinity. If true, this is astonishing intellectual achievement.

However, the existence of higher infinities is, on the face of it, absurd, for one very simple reason: infinity, by definition, is bigger than anything, and therefore there cannot be anything bigger than it. And we cannot practically count up to infinity. So what’s the use of postulating abstract structures that we cannot possibly construct?

Cantor’s reasoning is highly abstract. Can we trust it? And, even if we can, does it matter? Does it make any practical difference to our lives? Are there material consequences? Perhaps Cantor’s higher infinities identify deep, hidden structures that we’ve yet to fully interact with or notice in our empirical stream of experience? Or perhaps Cantor’s higher infinities are like the medieval proofs for the existence of God? We might grant that Cantor’s argument has some kind of logical necessity, but the premises just don’t connect to the reality we actually live in.

In this talk I give my (personal) conclusion on the status of higher infinities. I point out that, surprisingly, Cantor’s theory does entail some empirical predictions. But currently there’s no evidence to support those predictions (and quite a lot of evidence that suggests material reality prevents them).

But reality has a way of surprising us. We shouldn’t dismiss the importance of inductive creative leaps. So the jury is still out. A materialist attitude also includes the cheerful acceptance of ignorance. Sometimes we simply don’t yet know!

Cantor’s ideas raise questions about what rules of logical reasoning we are willing to accept. So they raise very profound and foundational issues about the identity of thought and being.

The fun part: you too will be learn to count beyond infinity (with the help of this handout: https://ianwrightsite.wordpress.com/wp-content/uploads/2017/02/infinityhandout.pdf). One of the surprising aspects of Cantor’s proof of higher infinities is its elementary nature. With just a small amount of effort, anyone can understand it.

30 minutes talk.