Computable (real) numbers are a countable subset of the reals and represent the largest set we can analytically reason about; however, equality of computable nu

Computable (real) numbers are a countable subset of the reals and represent the largest set we can analytically reason about; however, equality of computable numbers is not itself computable-- you can have computable c1 and c2 where cmp(c1, c2) is nonterminating (which means that c1 == c2 but it cannot be shown to be so in finite time). I'd argue that, for most people's tastes, computable analysis is wonkier than real analysis, even if the latter has uncountable infinities (which is rarely considered a problem).