How many groups are there up to isomorphism?
(I know it is infinite I want to know if it is countable).
What about rings and fields?
No, it isn't countable. Consider the groups $\displaystyle \bigoplus_{p_i \in S}C_{p_i}$, direct products of cyclic groups, where $\displaystyle S$ runs over all possible subsets of the set of prime numbers. There are uncountably many of these groups, as the set of subsets of an infinite countable set is uncountable, and no two of them are isomorphic.