On Mon, 12 Oct 2020 at 14:51, Stephen J. Turnbull turnbull.stephen.fw@u.tsukuba.ac.jp wrote:
As far as what Steven discussed, the ordinal numbers have the same properties (except I've never heard of ω-1 in a discussion of ordinals, but it should work I think).
I don't think it does. The ordinals are based on the idea of *orderings* of (potentially infinite) sets. So ω+1 is the ordinal of something like
1, 2, 3, ... 1
Addition is basically "bunging the second sequence at the end of the first". There's no obvious meaning for subtraction in the general sense here - you can't take a chunk off the end of an infinite sequence. And in particular, I can't think of an ordering that would map to ω-1 - it would have to be an ordering that, when you added a single item after it, would be equivalent to ω, which has no "end", so where did that item you added go?
(Apologies for the informal explanations, my set theory and logic courses were many years ago, and while my pedantry cries out for precision, my laziness prevents me from looking up the specifics :-))
Paul