Cardinality of the set of all numbers that modern math can define?

Question

I have recently learned that the algebraic numbers are countably infinite, and that very few transcendental numbers are known. Are enough transcendentals known to make up an uncountable set, or is the set of all numbers currently definable by mathematics countable?

Answer 1

If by definable you mean "explicitly constructible by chaining a finite string of logical symbols" then clearly the size of the set of all definable numbers is countable. In particular, this means that most transcendental numbers can't be written down explicitly as formulas.

However, if we allow for second order definability, then we can reach higher cardinalities.

An example of this is the completeness axiom for real numbers. We say that "every bounded set of real numbers has a least upper bound", but there's no explicit way inside the theory of real numbers for a variable to range over 'sets', only over objects, i.e. numbers. That's where second order variables come in.

Edit: Let me add some clarifications:

First: it's a somewhat disappointing fact of life that even though we're able to talk about sets of uncountable size, any kind of finitistic reasoning will only be able to pick out a countable part of our domain. That is, we'll only have enough tools to pinpoint exactly a very small fraction of the objects we create.

However, it's still possible to construct the reamining elements if we allow for non terminating algorithms.

For instance, to generate a undefinable number just set up a computer program that enumerates all definable numbers by ranging over formulas. Then, by following cantor's diagonal method, we can find the first few digits of our mysterious number. The trouble is this algorithm will never terminate, but still, we can see it's converging to something, and for the purposes of analysis, that's enough.

Answer 2

My comment might be viewed as facetious, but it really wasn't intended that way. The point is that the nature of what's accepted as a definition of a "number" really has a lot to do with the answer to this question.

If we accept this:

"The set of all real numbers between $1$ and $2$."

as an adequate characterisation of every real number in that interval, then clearly, the answer is that even that subset of the reals is uncountably infinite.

If, on the other hand, we try something like this:

"A "defined" real number is a real number with a finite explicit representation, i.e. one that is explicitly specified by a finite number of mathematical symbols and operations."

then the answer is probably "no". What I wrote is still horribly imprecise as a definition, but it's the best I can do. But the motivation behind it is clear. Transcendentals may be defined by such closed formulas, but if you enumerate all "possible" closed formulas, you'll still get a countable number. If that is how you view "definability", then your answer is "no, they're not countable".