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Comments on Is it known whether most numbers are normal or not

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Is it known whether most numbers are normal or not

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In a comment over at mathematics.stackexchange.com, I just (well, now stuff has happened and it's 8 hours ago) claimed that for most numbers it is not known whether they are normal. And then I got to think whether that is actually true?

Is there some set of numbers that (in some way, whether my measure, cardinality or something I haven't thought of) encompasses "most numbers", for which we know none (or just a few - in some meaningful sense) are normal? I believe that e.g. most rational numbers (but neither by measure nor by cardinality can they be said to be "most numbers") are not normal.

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the set of normal numbers in $[0,1]$ is measure 1. Which means the non-normal numbers are a null set.... (3 comments)
the set of normal numbers in $[0,1]$ is measure 1. Which means the non-normal numbers are a null set....
ziggurism‭ wrote 5 months ago

the set of normal numbers in $[0,1]$ is measure 1. Which means the non-normal numbers are a null set. Of course, contradictorily, the set of numbers which we can prove to be normal is probably countable.

Grove‭ wrote 5 months ago

So as I suspected my claim was (technically) wrong, but because most numbers are normal, not (as I thought) because most numbers aren't.

r~~‭ wrote 5 months ago

Your claim is correct as written, I think.

It is the case that for most numbers, it is unknown if the number is normal. (That is, for a real number between 0 and 1 chosen uniformly, the probability that we know whether it is normal or not is 0.)

It is also the case that we know that most numbers are normal. (That is, for a real number between 0 and 1 chosen uniformly, the probability that we have selected a normal number is 1.)

English is not great at highlighting the difference between the first claim and the negation of the second!