You have the natural numbers, then the integers, then the rational numbers, then the real numbers, which is the set of rational and irrational numbers. But are there two distinct sets of irrational numbers? Some irrational numbers like e or pi or the square root of 2 are derivable. But it’s also possible to have a random irrational number. Is there a term for the set of numbers that includes the rational numbers and derivable irrational numbers but doesn’t include random irrational numbers?

I think that’d be the computable numbers, if I understood you correctly.

Or maybe the division between algebraic numbers and transcendental numbers.

As an aside, the cardinality of the set of computable numbers is still only the same as the cardinality of the integers. Which means that even that very large (for human purposes) set is still only an infinitesimally small portion of the full set of real numbers.

EDIT: The OP is not referring to algebraic numbers, since he uses e and pi as examples, both of which are computable but transcendental.

Unanswerable until the OP defines what he means by “derivable”.

The most familiar hierarchy of fields beyond the rational numbers is as follows:
[ul]
[li]The constructible numbers: These are the points on the real line that you can get to from zero and one using a compass and a straightedge.[/li][li]The algebraic numbers: These are the roots of polynomials with integer coefficients.[/li][li]The computable numbers: These are the reals that can be approximated to any degree of accuracy with rational computations.[/li][li]The definable numbers: These are the reals that can be given finite-length descriptions.[/li][/ul]
You could make the list a lot longer, but nothing I’ve left off has a particularly natural characterization. Note that all of these are countable sets, and each one is a proper subset of the next one.

That said, this doesn’t have quite the same feel as the distinction between the more familiar number systems. In the standard construction of the real numbers, we start with the axioms of set theory, use those to define the naturals, and then use simple set operations to get the integers, the rationals and the reals in order. You probably could stick in some more complicated constructions to get the others above, but it’s not clear what that would buy you beyond a much more complicated theory.

ETA:

Either computable or definable, which most people will see as equivalent notions unless they have some pretty specialized knowledge.

The example I’ve seen to distinguish computable from definable is the halting Omega, a number which encodes whether a given Turing machine will halt on any given program. So long as the Turing machine is fully specified, Omega can be defined, but since a computation of Omega would require a solution of the halting problem, it cannot be computed.

That’s a pretty botched explanation. Chaitin’s constant Omega is the probability that a randomly chosen program will halt when given a randomly chosen input. Not only is it not computable, but you can’t say anything about its value without assuming the answer.

For purposes of this thread, let’s say a derivable number is one that two mathematicians could determine independently of each other by starting from a common definition. If, for example, two mathematicians were each determining the square root of two to five hundred places, they’d come up with the same number. This wouldn’t happen with a random number.

This sounds like what I was looking for.

A very interesting set of numbers is Erret Bishop’s constructible number (not to be confused with the geometrically constructible numbers). One of these numbers is defined by two recursively enumerable sequences (i.e. sequences that can be the output of Turing machine) of integers, say {a_n} and {b_n} with the property that it is provable (constructively, of course) that a_n/b_n - a_m/b_m < 1/n + 1/m for all n and m. Then the limit a_n/b_n is a constructible number. It is interesting that there are constructible numbers in this sense for which, in our present state of ignorance, we cannot write down a single decimal place. Hint: we might know, for example that it starts out either .99999… for an incredibly large number of places or that it starts out 1.00000…, ditto. It might be exactly 1, or differ from 1 by an incredibly tiny number and we don’t which.

Hmm, not sure if there’s some misunderstanding – Omega is the probability that a given (prefix-free) Turing machine will halt given a random program; and while it’s not computable, it’s left-computably enumerable, i.e. the set of rational numbers that are smaller than Omega is computably enumerable, and thus, you can, in principle, construct better and better approximations to its value (as Calude has done, for example).