Integers (symbol Z) are countable as well. The set of integers contains zero and all natural numbers as well as their negatives. To count them all, just start with 0, then 1 and -1, then 2 and -2, etc.

Rational numbers (symbol Q) are those numbers that are expressible as fractions, e.g. 1/2. Rational numbers are also countable. For an idea of how to count them, imagine a two-dimensional grid that has integers on each axis, and contains infinitely many horizontal and vertical lines, one for each integer on the respective axis. To count all rational numbers, start at (0,0). Then, count all intersections on the border of the square between (-1,-1) and (1,1). Then, count all intersections on the border of the larger square between (-2,-2) and (2,2). And so on.

Algebraic numbers are those numbers that are roots of a polynomial. Algebraic numbers include e.g. the square root of 2, which is the root if (x^2 - 2 = 0). Algebraic numbers aren't necessarily rational, but they are also countable. They are countable because their quantity is limited by the quantity of polynomials, and polynomials are countable. (A polynomial is determined by its coefficients, and the coefficients can be written down as a number.)

Finally, we arrive at real numbers. Real numbers (symbol R) are an expansion of rational (and algebraic) numbers that is intended to address a certain deficiency. The deficiency is that there exist convergent series of rational/algebraic numbers which appear to converge to

*something*, but that something is not a rational (or algebraic) number. A classic example of such a convergence is Pi.

The idea behind real numbers is to have a set which, in addition to rational numbers, also contains the converging points of all converging series that can be constructed within the set. This makes the set "complete".

The traditional idea is to envision a real number as an infinitely long series of zeroes and ones (i.e. the number itself expressed in binary, with a decimal point some place).

If one imagines a real number like that, then it intuitively follows that a set of all-real-numbers is the set of all such infinite-series-of-0-and-1.

Cantor's diagonal argument can then be used to show that a set defined that way is uncountable. If you have an infinite set of unique infinite-series-of-0-and-1, such that each series in the set is marked with an index; then you can construct another infinite-series-of-0-and-1 which differs from all the other series that have an index. So therefore, the indexing is not complete. The set contains more series than can be counted.

Well yes, but: if we are constructing the set of real numbers as the set of rational numbers + the converging points of all converging series, then aren't the converging series countable? Every converging series must be the result of some algorithm. But algorithms are expressible as code, and code is expressible as numbers. So the number of possible algorithms is countable. So the number of all converging series must also be countable. If we construct the set of real numbers this way, then, it is a union of two countable sets - so it must itself be countable.

Does it not follow, then, that the standard envisioning of real numbers as infinite-series-of-0-and-1 must include an infinite number of them that are neither rational, nor the result of any limit? If so, what is the use of all these extra, unidentifiable "real numbers"?

It sure seems that, all those real numbers that we can actually express, or use in any way - are countable.

Or is the number of converging series uncountable? If so - how?