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Comment by thriftwy | Hacker News Reader

I don't agree. What if the next symbol's meaning conditionally depends on the previous ones?

For example, in 1937 the third glyph is number three, but in ВАЗ, the third glyph is Cyrillic letter Z. They're not different in writing.

It would not be hard to devise a symbolic system where new symbols are context dependent, and there are infinitely many of these depending on the context. The context will be governed by a finite number of rules. The glyphs will not change meaning in a specific text.

Human language is an example of such system. If you treat words as a whole as symbols, you can have infinitely many different words as they become defined via the specific text, but obviously you can't have infinitely many different words in a single text because they will stop being mutually intelligible at some point.

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Of course you can have infinitely many different semantics, but this is a question about available syntaxes. (Trivial proof: for each n, take the semantics S_n which assigns to the symbol "X" the meaning "integer n", just as English usually assigns to the symbol "3" the meaning "integer 3".)

I'm not sure I follow. Natural languages usually have very simple syntax. The Greeks had great literature while still in "boustrophedon with no whitespace" phase.

The number of possible symbol meanings may be unbound. And there may be symbol definition rules so that any effective symbol set used to write a text is still plausibly an alphabet. Chinese actually does fit the bill if you are allowed to define new valid characters according to best practices.

You're talking entirely about the meaning of symbols, right. That's not a question Turing addressed at all, and the answer is trivially "there are infinitely many possible semantics". Turing only addressed the question of how many physically distinct symbols there could be, and came up with the result "finitely many".

The Turing machine has a memory, it can absolutely cope with symbols meaning something different depending on what appears before them. The trouble is that it can't distinguish between infinitely many distinct symbols on the paper tape. Turing here shows that a human can't do that either, so that is not a way in which humans have more computational power than a Turing machine.

Not being to able to distinguish infinitely many symbols on a single tape does not mean there is a bound on the number of meanings that may be assigned to symbols.

The alphabet itself would still be finite, like our vocabulary.

Our vocabulary is only statistically finite. Number of words in a dictionary is finite. Number of words used to all speakers is not.

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