Definition talk:Counting Measure

Is this the same thing as the Definition:Cardinality of a set, or is it subtly different?

The problem I have with this is that the definition of natural numbers appears to be defined in terms of themselves. In order to define the order of the sets $$n+1 = o\left({N_n}\right) \ $$, you are already talking of a map that takes the set to the "number of elements it has", and hence the definition which ultimately seems to mean: "A number is the number associated with that number".

I haven't done yet, I mean to get round to doing, the construction from Definition:ZF, I just wanted to get my hands on a copy of Bourbaki first. This I have done, but it's going to take me a while to decipher it into a form where lubbers like me can explain it simply. That is, by defining $$0 = \varnothing, 1 = \left\{{\varnothing}\right\}, 2 = \left\{{\varnothing, \left\{{\varnothing}\right\}}\right\}, \ldots$$ and hence (from the Axiom:Axiom of Infinity) $$N + 1 = N \cup \left\{{N}\right\}$$. Thence to show, via the concept of the Definition:Naturally Ordered Semigroup, that the set fulfills the Peano Axioms and so on and so forth. (But it's so far incomplete and I'm bogged down.) --Matt Westwood 22:40, 20 January 2009 (UTC)


 * I think it is subtly different, in that the counting measure of ANY infinite set is undefined or just said to be "infinite", where as the different levels of Cantor's infinities are well-understood as cardinality.
 * The construction I was going for here is precisely what you're referring to. I may have made an error in my presentation of it (in fact, I probably did, as you point out) and we can refine it once we get those materials to make it precise, or just delete it entirely, since I doubt it's used in any proofs.  Zelmerszoetrop 22:55, 20 January 2009 (UTC)