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A little diagram ive been tinkering with to show the relationship between the different types of Real Numbers.
@freemo, nice!
Another way to categorize numbers is according the information they store.
Natural numbers are infinite, but each number store a finite amount of information.
I don't know if computable numbers with infinite digits stores finite or infinite information, but for sure each of them can be compressed to a finite representation, i.e. the algorithm computing it.
Instead real numbers with infinite arbitrary digits, have infinite information, because there is no finite algorithm that can list all the digits of one of this numbers.
Probably these real numbers are not part of the universe, because they store infinite information.
The real numbers with arbitrary infinite digits can be built in math, thanks to the power-set axiom, when you build the set of all sets of an infinite set. This is a very big set, and you can pick all elements from it, thanks to the axiom of choice.
> I'd be reluctant to equate "infinite digits" with "infinite information"..
yes, I see your point. For answering in a meaningful way, I should study Information Theory (https://en.wikipedia.org/wiki/Information_theory), entropy, and so on, but I have no time/energy 🙂 And probably after doing this, your point remain valid!
So I downgrade my affirmation to: it is funny noting that any number in your list can be represented with a finite notation, except a number of type "uncomputable real", which requires an infinite number of symbols for being represented. There is no possible notation that can represent an "uncomputable real" using a finite number of symbols.
Many of them can not even be defined with a finite number of words, because they are an infinite sequence of arbitrary chosen digits.
In the linked video, they say that uncomputable reals are the majority of numbers, but I note that they are probably not interesting because you can not associate them to nothing that exists in this universe.
@mzan yea uncomputable reals are the majority of numbers, but while i wouldnt say they are less interesting, they are, right now, less functionally useful since we cant compute them in the first place. So while I'd say they are terribly interesting, they simply arent terribly useful :)
@freemo ,
> So while I'd say they are terribly interesting, they simply arent terribly useful :)
Don't let me wrong, you are not the typical useful and popular type of number that one usually note while he is doing math, and which is reported on all mathematical papers. But I'm sure that you will find a mathematician that will find you interesting, and who will appreciate you! Good luck my uncomputable not existing real number ! :-)
@mzan I'd be reluctant to equate "infinite digits" with "infinite information".. a random number generator can produce infinite digits but has 0 information. Most numbers that have infinite digits after the decimal point, wont have any more information than a random number generator, well no more so than an integer anyway.