> Something countably infinite proceeds towards infinity in one direction. This is a mental trap when thinking about these things. In fact the rationals are al

> Something countably infinite proceeds towards infinity in one direction. This is a mental trap when thinking about these things. In fact the rationals are also countably infinite (a/b, with a and b integers), as are the "complex integers", that is, the set (a + bi) with a and b integers, and i = sqrt(-1). The "2-dimensional" set of tuples (x, y) with x and y integers is countably infinite; in fact any n-dimensional set of tuples of integers is also, for finite n. Many more things are countable than most people realize. > The thing about real languages though is that there is an infinite number of possible interrelations between any two words based on context. The set of all possible books is countably infinite. Think about that for a bit. If your "context" can be encoded in any number of millions of books, encyclopedias, and dictionaries, then I can simply append that context onto whatever else I'm saying, and now I have my text plus all context. Even if you say "but books can't capture the subtle tonalities and facial expressions of human expression", you still have to realize that we have finite photorecptors in our eyes, and finite hairs in our ears, and finite neurons to process all that information. So the set of signals a human could ever possibly process as distinct must be (at most) countably infinite as well. You cannot get uncountably infinite language without infinitely large brains. I know we want to all feel like we're magic and special, and this mysterious uncountable infinity feels like it leaves lots of room for us to have magical consciousness and a soul and an afterlife and X-Men superpowers, but it just isn't there. It just doesn't work. Besides, don't underestimate the size of countable infinity. That's not exactly something to get claustrophobic about.