This may not strictly be a "note for nowhere else" but I am test running it here first because of the frequency with which assertions turn up on
The importance is that even world renouned mathematicians get this wrong way to often. I just finally started reading Roger Penrose's Shadows and quickly found him confounding the number of algorithms with a countable number (even though he had just made mention of Cantor and "diagonal" in the same breath) in his effort to make a case that there are things human minds can do that are not computable. Meanwhile, in trying to make the opposite case, Wolfram cites the "oracle" argument which contains exactly the same misunderstanding.
While there are a whole lot of technical arguments that can be got into about orders of infinity, there is really only one distinction that matters to our everyday understanding of the world, that between aleph 0, countable infinity, and aleph 1 (and higher) which is uncountable.
I could just make the point that actualities are countable and possibilities are uncountable, but that would be putting the cart before the hourse.
First we need to recognise that any form of infinity is just a mathematical construct. We are not ever going to have to deal with actual infinities, but what is very useful is understanding limiting behaviour as something or other goes towards infinity. It is here that the difference between countable and uncountable infinities matters immensely.
- can be put into a one to one correspondence with the natural numbers
- cannot be put into a one to one correspondence with the natural numbers
The digits of pi, the keystrokes made by a room full of monkeys, the galaxies, stars, grains of sand, atoms are all countable. Each could be given a number.
Possible texts, possible algrithms, possible macromolecules, possible species, potential humans cannot be given a number. Each is a unique combination of a long sequence of units and every time you add one to the length of the base sequence, you multiply the possibilities by some factor greater than one, often much greater but even that does not matter except in the limit.
So if you struggle to count the possibilities of a small complex system, just making that system slightly larger multiplies those possibilities out of all proportion and counting them becomes orders of magnitude harder.
Even if, as I strongly suspect, our cosmos is but a semen stain on the sheets of some grand cosmological process going on beyond our event horizons, there is no "other earth" anywhere else. There is no other you. There is no other me! Fortunately!!