> A mathematician named Kantor Cantor. Georg Cantor. > And that kind of infinity is called, well, countable or aleph sub(0). It's like O(n) in algorithms. It

> A mathematician named Kantor Cantor. Georg Cantor. > And that kind of infinity is called, well, countable or aleph sub(0). It's like O(n) in algorithms. It's not really at all connected to O(n), and only tenuously connected to Big-O notation at all. Big-O notation works over integers or reals, or even some other (possibly finite) sets. It doesn't make sense to say O(n) is countable any more than it makes sense to say that the line "y = 2x + 7" is countable. What would that mean? Especially if x and y are real numbers? > our favorite transcendental numbers, you know them, pi, e, psi, that lot, are not part of that. Neither are multiples, or fractions, of those numbers. True for pi and e, but what is psi? Do you mean phi, the golden ratio? Or do you really mean psi, the sum of the reciprocals of the Fibonacci numbers (I had to look this one up)? The golden ratio (phi) is not transcendental: phi * (1 - sqrt(5)) is -2. It doesn't look like it's known whether psi is transcendental or not. > At any rate, possible physical distances are uncountable. There's no particular reason to believe this is true, and some reason to believe it's not. Look up the "Planck length"; below this length it's not clear whether the concept of "distance" is even meaningful. > And living things with brains have a (probably) countable number of neuronal interconnections ... Not just countable neuronal interconnections: literally finite. Neurons have finite size and your brain isn't infinitely large (sorry). > ... each of which depends on uncountable physical distances. Pseudoscientific mumbo jumbo. Not even wrong. Literal nonsense. > My question: is this digital infinity countably infinite? Or does it go beyond that? It is countably infinite by definition. It's isomorphic to the free monoid over the (finite) digits. > Does this countability matter to our understanding? No. Uncountability is a curious feature of our model of real numbers. All models are wrong, but some models are useful. There's no real evidence that the uncountability of reals is an actual useful feature of that model, and not just a curious edge-case artifact. Most likely there is no physical analogue to uncountably infinite sets (my opinion, obviously). Am I nitpicking you? Details matter. You seem pretty careless with your facts here, which is a great way to accidentally spread disinformation. Maybe try to be more careful in the future.