[OskarS]

You've misunderstood the definition of normal numbers. A "normal number" (loosely defined) is a number where the digit expansion of the number has all possible substrings of digits uniformly distributed in the limit as it goes to infinity.

Several things to note:

1. Whether or not a number is normal has nothing to do with the "randomness" of its digits or the "amount of information" in the digits. Chapernowne's constant, 0.123456789101112... isn't "random" at all and contains very little information, yet it is known to be normal in base 10.

2. A number can have totally super-random digits (as random as you want! it can even be non-computable! infinite information!) and not be normal at all. For instance, imagine you have a normal number that's as random as you want, and it starts like:

0.892345123402345671235....

And then goes on forever, no discernible pattern. Say you construct a new number from this number, with the only difference being that you remove the digit 7. Literally, every place that a 7 appears in your original number, you just remove it. This number would no longer be a normal number, because all the sequences with the number 7 in it would appear nowhere.

The number would still have "infinite information" in the sense of the author of this paper. It would still be "just as random", it would still have "no pattern". But it would not be a normal number anymore.

Whether or not a number is "normal" or not has nothing to do with the issues raised in this paper. "Normality" is a different criterion entirely. When the author talks about numbers with "infinite information", he's not talking about normal numbers, he's talking about computable numbers, which is an entirely different concept: https://en.wikipedia.org/wiki/Computable_number

[throwawaymath]

> Chapernowne's constant, 0.123456789101112... isn't "random" at all and contains very little information, yet it is known to be normal in base 10.

Why is Chapernowne's constant not random?

> The number would still have "infinite information" in the sense of the author of this paper. It would still be "just as random", it would still have "no pattern". But it would not be a normal number anymore.

That's not true. It would not be just as random. If the set you're sampling from includes a 7 (i.e. the set of digits which can be represented in any single place in the sequence), and you never see a 7 for an extremely long time, this is exceptionally good heuristic evidence that the number is not random. And if we know 7 never shows up, we also know that the number is not random, because we know it's not uniformly sampling from the set of base 10 digits.

[sykh]

I read through most of the paper we are all nominally talking about. He doesn't use normal numbers as you say. He is talking about computable numbers as you say. My apologies.

But a normal number has "random" digits in the sense that all finite sequences of digits (in a given base) occurs uniformly in the expansion. What other notion of random can one meaningfully give for an expansion of a number's digits? Without getting into too much philosophy.

From [1]: We call a real number b-normal if, qualitatively speaking, its base-b digits are “truly random.

My expertise is in commutative algebra so I'm outside of my comfort zone.

[1]: https://www.davidhbailey.com/dhbpapers/bcnormal.pdf

[OskarS]

Something can be random and not have a uniform distribution. If I throw two dice, the sum will be random, but it will not be uniformly distributed. Uniform is just one distribution among meny.

The only definition of "randomness" in a sequence that holds any kind of water, philosophically or mathematically, is Kolmogorov complexity [0] (see specifically the section on "Kolmogorov randomness"). I don't know where you got the idea that "normalness" is the ultimate version of randomness, but it's not.

Forget the formal definition for a second, and just think of the intuitive notion of randomness for a second: does the sequence 12345678.... look random to you? In random sequences there should be no patterns: do you see a pattern in this sequence? If you had a computer program with a random number generator that produced that sequence of digits, would you be happy with it? No, you wouldn't.

In the sense of Kolmogorov randomness, the sequence 1234567.... is obviously not random at all, since it's trivial to find a Turing machine to generate it. It matches up perfectly with out intuitive notion of what randomness is, and it quite correctly points out that the amount of information in the string is very low, even though it's infinitely long. That is my definition of randomness, and it's (more or less) the author of this paper's definition.

[0]: https://en.wikipedia.org/wiki/Kolmogorov_complexity