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Just a reading note to myself: the key assumption is that there is a "resolving limit" \epsilon such that symbols that are "closer and smaller" than \epsilon ar

Just a reading note to myself: the key assumption is that there is a "resolving limit" \epsilon such that symbols that are "closer and smaller" than \epsilon are indistinguishable. The implicit assumption there is that you can resize symbols while keeping them the same.

andrewflnr 2y ago
I'm not sure that assumption is necessary, at least for finitely sized alphabets. For finitely many sizes of symbol that are meant to be distinguishable, you can define the smaller ones as just taking up smaller portions of the square. If you try to scale down too far, they'll eventually hit the epsilon limit and be indistinguishable again. Maybe you could break the theorem by growing arbitrarily large, but that really stretches the word "alphabet". He was thinking of notebook pages, so encoding things in the size of arbitrarily large symbols wasn't top of mind.
- contravariant 2y ago
  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

I'm not sure that assumption is necessary, at least for finitely sized alphabets. For finitely many sizes of symbol that are meant to be distinguishable, you can define the smaller ones as just taking up smaller portions of the square. If you try to scale down too far, they'll eventually hit the epsilon limit and be indistinguishable again. Maybe you could break the theorem by growing arbitrarily large, but that really stretches the word "alphabet". He was thinking of notebook pages, so encoding things in the size of arbitrarily large symbols wasn't top of mind.

contravariant 2y ago
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.