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I dunno, somehow the conclusion that an infinite alphabet requires symbols to grow arbitrarily large to allow them to be distinguished from the others doesn't s

I dunno, somehow the conclusion that an infinite alphabet requires symbols to grow arbitrarily large to allow them to be distinguished from the others doesn't seem quite as interesting as the conclusion that all written alphabets are finite. I mean clearly you can design an infinite alphabet by just using numbers for all symbols and writing the numbers in whatever system you want. Sure some symbols will require more paper or their details become too small to see, but it is still an infinite written alphabet. The more obvious argument is that an infinite alphabet is pointless because nobody can remember an infinite list of symbols (hence any practical example of such an alphabet must have some rules to reduce the symbols to a finite set of atoms). There's nothing theoretically impossible about one existing.

andrewflnr 2y ago
> somehow the conclusion that an infinite alphabet requires symbols to grow arbitrarily large to allow them to be distinguished from the others doesn't seem quite as interesting as the conclusion that all written alphabets are finite. Well yeah, I only brought that up to try to steelman the comment I was responding to. > I mean clearly you can design an infinite alphabet by just using numbers for all symbols and writing the numbers in whatever system you want. Sure some symbols will require more paper or their details become too small to see, but it is still an infinite written alphabet. That's all explicitly covered in the article. In particular, not having details too small to see is basically the definition of an alphabet for the purposes of the theorem. And as soon as you're using "numbers", you aren't really using an "infinite alphabet", you're using whichever numeral system with an extra layer of interpretation. Turing separately covered the case of simulating an infinite alphabet with countably infinite strings in a finite alphabet. Ed: clarity.

andrewflnr 2y ago

> somehow the conclusion that an infinite alphabet requires symbols to grow arbitrarily large to allow them to be distinguished from the others doesn't seem quite as interesting as the conclusion that all written alphabets are finite. Well yeah, I only brought that up to try to steelman the comment I was responding to. > I mean clearly you can design an infinite alphabet by just using numbers for all symbols and writing the numbers in whatever system you want. Sure some symbols will require more paper or their details become too small to see, but it is still an infinite written alphabet. That's all explicitly covered in the article. In particular, not having details too small to see is basically the definition of an alphabet for the purposes of the theorem. And as soon as you're using "numbers", you aren't really using an "infinite alphabet", you're using whichever numeral system with an extra layer of interpretation. Turing separately covered the case of simulating an infinite alphabet with countably infinite strings in a finite alphabet. Ed: clarity.