Doron,
If you are still reading this thread, you'd asked about a relationship between N and |N|. I will now provide my answer.
There are of course many, many ways in which the set of natural numbers and its cardinality are related; there are also many, many ways in which the two are unrelated. (And that's why what you asked was not a well-formed question.)
Be that as it may, they are related in a way I find interesting, but in order to present it, I must take an aside to lay my foundation so we are all on the same page. I'll develop the set of integers in a fairly standard set-theoretic way.
Define 0 to be {} and define the Successor function, S(x) to be (x∪{x}).
S(0) = (0∪{0}) = ({}∪{0}) = {0}. The successor of 0 we will call 1. So, 1 = {0}.
And then,
2 = S(1) = (1∪{1}) = ({0}∪{1}) = {0,1}
3 = S(2) = (2∪{2}) = ({0,1}∪{2}) = {0,1,2}
4 = S(3) = (3∪{3}) = ({0,1,2}∪{3}) = {0,1,2,3}
So, ∀x∊N, x = {0,1,2,...,x-1}. That is, any natural number x is the set of all natural numbers less than x.
So far so good? All I have done is represented the natural numbers as sets.
The ZF Axiom of Infinity reads as follows: ∃N ({}∊N ∧ ∀x∊N ((x∪{x})∊N))
It postulates the existence of a set that includes the empty set and the successor for every member of the set. This is precisely the set of all natural numbers.
What about cardinality? Well, take 6 for example: 6 = {0,1,2,3,4,5} and |{0,1,2,3,4,5}| = 6. So, |6| = 6.
As it turns out, ∀x∊N, |x| = x, and by extension, |N| = N.
In other words, N = [font=+4]ℵ[/font]0
Well, I find it interesting, at least.
