AFAIK pi, e, psi, etc are countable. I mean, you can count them: pi is 1, e is 2, psi 3, etc. If you invent a new transcendental real number that'll be 4. etc.

AFAIK pi, e, psi, etc are countable. I mean, you can count them: pi is 1, e is 2, psi 3, etc. If you invent a new transcendental real number that'll be 4. etc. Adding all integer multiples and fractions is also countable, etc. Uncountability starts at the uncomputable reals: these are numbers that only exist as infinite collections of digits and don't have a more concise definition. This includes things like Chaitin's Constant, which is the probability that a randomly-generated Turing Machine will halt. The Cantor diagonalization argument also worked with uncomputable reals - again, infinite sequences of digits. Computable reals are countable. Physical distances are not uncountable. At the very least we know space cannot be infinitely subdivided: resolving any distance shorter than a Planck length creates black holes. So that would limit us to countable infinities. The digital infinity is likely not even infinite; just "way too big a state space for humanity to ever feasibly exhaust absent concerted effort to do so". Though, if you want a story that plays around with this, check out Melancholy Elephants[0]. [0] http://www.spiderrobinson.com/melancholyelephants.html http://www.spiderrobinson.com/melancholyelephants.html