How real are real numbers? (2004)

You didn't prove that statement, and actually Cantor's diagonal arguments [1] proves you wrong:

Consider the set of «all real numbers which you can actually specify IN ANY FASHION AT ALL». If you can count it, you can order them in a certain fashion: n0, n1, n2, n3 … It's easy to specify a number X which is not part of this set (which is in contradiction with the definition of this set, and demonstrates ad absurdum that this set is not countable). To build such X, consider the n-th decimal of the n-th number: if it's zero, the n-th decimal of X is 1, otherwise the n-th decimal is 0. Then by construction, X is different from any number of this set, which mean X is not a part of the set. □

[1]: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

First of all, the set of values in the range [0,1] that can be represented by a computer is not at all the same as the set of rationals. If you're talking about fixed-point or floating-point data types, then there are only a finite number of bit patterns (corresponding to dyadic rationals), so obviously the probability of hitting one of them precisely is zero.

You may think I'm just being pedantic here, but I'm really not. The theorem you described is only interesting because there are an infinite number of rationals between zero and one, and yet the set still has measure zero. But if you're willing to say that any rational number is theoretically "representable" in a computer, then you're already assuming that we can go beyond the natively-available data types, and come up with our own representations -- e.g., a tuple of numerator and denominator. And in that case, we can also come up with representations of irrational numbers; for example, you could have a datatype that can finitely specify any algebraic number, such as sqrt(2) or the golden ratio. So the fact that a typical CPU happens to only be able to natively deal with rationals is irrelevant.

The beauty of the diagonalization argument described in this paper is that it applies to all possible discrete representations, no matter how weird or complex, without having to make any assumptions.

The way you defined that number makes it a perfectly valid element of the set R, as described by, say, the axiomatic definition here:

https://en.wikipedia.org/wiki/Real_number#Axiomatic_approach

Whether it's easy or hard or computationally intractable to compare it to other numbers, that's a totally different question unrelated to its definition.

Plus, you can actually empirically compute a finite set of initial digits (a specific Turing machine can be analyzed to see if it terminates or not), so you can compare this number with one that's constructed by flipping its digits, or with pi, etc.

In order to draw a rational number, you must randomly select a number that has a repeating pattern in decimal. Imagine that for the sake of argument that we're in this "repeating pattern" and, amazingly, we've already drawn it 1000 times! What a roll we're on! In order for the number to be rational, how many more times must we draw this pattern? Infinitely many. What is the probability that in one of those infinite fair random draws, the repeating pattern gets broken? 1. It's just not possible for infinitely many fair draws to produce a repeating pattern; that would be proof that the process generating the number was in fact not random in the first place. (I can't be that sloppy if we're doing finitely many draws but I believe that's justifiable at infinity.)

Imagining through this process of drawing a random number I think makes this more intuitively obvious than imagining being presented with a completed number and trying to figure out if it's rational.