Fun fact: the set of algebraic numbers is countable. If you recall that the set of rational numbers is also countable, then you get that the reals are uncountable only "because" of transcendental numbers (pi, e, Chapernowne's number, etc.).
As Alonzo Church and Alan Turing showed, the computable numbers [0] are countable too. The computable numbers include all the algebraic numbers and some transcendental numbers (including π and e), so the reals are uncountable "because" of those other transcendentals, the uncomputable numbers. Put differently, almost all reals are uncomputable.
This is really interesting. We could take it further and say that, given that some uncomputable reals have a finite definition (e.g. "the probability that a random algorithm halts"), there is a countable number of definable reals (by assigning a Godel number to each definition), so the uncountability of the reals is strictly due to indefinable numbers!
I said uncountably infinite. The parent seemed to be under the impression that a few specific transcendental numbers were the reason that the reals were uncountable. I was pointing out that there are uncountably infinite irrational numbers between any pair of rational numbers. (There being more irrational numbers than real transcendental numbers, as not all irrational numbers are transcendental.)
You are saying rational numbers are countable
but irrational numbers are not countable (uncountably infinite)
This is true.
The parent is saying that algebraic numbers are countable
and transcendental numbers are not countable.
The parent's statement is a level deeper and more remarkable.
Another way to say this, is that "almost all" numbers are transcendental.
The parent was not suggesting that his short list of examples represented the extent of transcendental numbers: despite almost all numbers being transcendental, they are individually hard for us to define, since the set is normally defined by excluding sets of numbers that we can easily define.
To clarify: transcendental numbers are a subset/a type of irrational numbers. https://en.wikipedia.org/wiki/Transcendental_number
If you take out transcendental numbers from the irrationals, those that remain are algebraic and therefore countable. You got it :)
The parent says, and I quote: "...then you get that the reals are uncountable only 'because' of transcendental numbers."
But this is untrue, is it not? Because even if you removed all of the real transcendentals, the real numbers would still be uncountable because of the irrationals.
(Edit: Or are the non-transcendental irrationals countable? I can see how that might be true?)
