So how does it "act like a number"?
Additional comment: I don't care what kind of infinity or infinite set you may define, I can count them one-for-one as you produce another number from it. Forever. So what's so deep about showing gaps in any proposed ordered counting system? None. Infinity is not a number. Any infinity can be counted by the natural numbers forever. It's smoke and mirrors to say any infinite set is larger than another (assuming, without accepting, that the term 'infinite set' actually has any meaning. Can a "set" be an unbounded amount? Possibly only in fantasy, which is what most discussion of infinities amounts to.

So how does it "act like a number"?

Additional comment: I don't care what kind of infinity or infinite set you may define, I can count them one-for-one as you produce another number from it. Forever. So what's so deep about showing gaps in any proposed ordered counting system? None. Infinity is not a number. Any infinity can be counted by the natural numbers forever. It's smoke and mirrors to say any infinite set is larger than another (assuming, without accepting, that the term 'infinite set' actually has any meaning. Can a "set" be an unbounded amount? Possibly only in fantasy, which is what most discussion of infinities amounts to.