(November 18, 2022 at 1:25 pm)Jehanne Wrote:(November 18, 2022 at 12:33 pm)polymath257 Wrote: I'm always leery about popular treatments of mathematics, especially treatments of infinity.
What would you say is the core of his argument?
He introduces ZFC and says that, apart from finitists, mathematicians no longer (since Cantor) view the concept of infinity as being equinumerous; as you know, there are an infinite number of infinite sets, one countable and an infinite number that are not. Applying infinite attributes to a God in a qualitative manner is a cop-out, and doing so in a quantitative manner is meaningless.
One aspect is that there are several, very distinct, notions of infinity. Cardinality is only one and probably the easiest one to describe to a layperson. But there are also ordinals (having to do with the order structure). In ZFC, every cardinal number is an ordinal, but not the reverse.
Also, not only are there an infinite 'number' of different sizes of infinite sets, the collection of all cardinalities is a proper class and not a set (like the class of all sets). But this gets into the paradoxes of naive set theory (as opposed to axiomatic set theory--given by ZFC).
