>>127
infinitesimals
While I can't say they sit too well with me, they are well defined, at least within set theory.
Here's an example:
x
2 = 2, where x > 0 => x = √2.
It can be shown that x is not a rational number (that is, you can prove that there is no p/q = x, with (p,q)=1). The proof is elementary and is shown in middleschool textbooks, so I'll refrain from showing it here.
One way of defining x is to use 2 infinite sets in ZFC:
{{y∈Q|y
2 < 2},{y∈Q|y
2 ≥ 2}}
This is called a Dedekind cut. You're defining all the rational numbers to the left (smaller) and to the right (greater or equal) to this irrational number. By doing this you can approximate this irrational number with ever-increasing accuracy. Would any side ever reach this number, despite approaching it ever-so-closely? No, because that would require infinite number of digits in the fraction, which is not something a rational number can be, however the cardinality of both of those sets is aleph-null (infinite, same as N) as they are an interval of Q.