Cardinality is the name given as a measure of the size of a set. The cardinality of the rationals is the same as the integers. The cardinality of the reals is larger than that of the integers.
There is an arithmetic of cardinal numbers and it is well understood if you accept the axiom of choice. For instance, using your notation, N*N = N and N^2 = N. You can read more here
There is an arithmetic of cardinal numbers and it is well understood if you accept the axiom of choice. For instance, using your notation, N*N = N and N^2 = N. You can read more here
https://en.m.wikipedia.org/wiki/Cardinal_number