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I'm still reading it, but I don't agree with the following: >In Plato and the Nerd, I review Claude Shannon's channel capacity theorem [Shannon, 1948], which s

by GTP 4y ago

I'm still reading it, but I don't agree with the following: >In Plato and the Nerd, I review Claude Shannon's channel capacity theorem [Shannon, 1948], which states that any noisy observation of anything conveys only a finite number of bits of information. Borel's know-it-all number cannot be encoded with finite number of bits unless the list all possible yes-no questions is finite, which it is not. It is easy to construct an infinite sequence of valid yes-no questions. For example, let the first question be "Is one a whole number?" Let the second question be "Is the answer to the first question YES?" Let the third question be "Is the answer to the second question YES?" And so on. As a consequence, Borel's number cannot reveal itself to us unless we invent some noiseless way of observing a thing in itself. It is true that he showed an example of an infinite list of valid yes/no questions, which would lead to a number with an infinite amount of digits. But despite it being infinite, by the way it was constructed you already know all the digits (they will be all 1s), meaning that that number's entropy is remarkably low and that number can be expressed with a finite number of bits of information. So Shannon's theorem isn't a problem here.

kirkules 4y ago
The presentation in the quote is awkward and doesn't work, as you pointed out, but Lee could instead have just mentioned that there's such a construction with questions of the form "is the answer to the previous question YES [NO]?" which is not constant (not all 1s), and does have the property he wants, i.e. the are such sequences that cannot be specified finitely (corresponding to binary sequences with that property). I don't think you've shown that Shannon's theorem isn't a problem here so much as the argument presented did not successfully convey that it's a problem. Tangentially, I'm interested to see a formal statement/proof of the form of Shannon's theorem add used in this article, because I can sort of intuitively see why it makes sense but I'm not familiar enough with the technical details to guarantee that this is even an appropriate use of the theorem, assuming the yes/no question construction has all the properties Lee wants it to have.
- pencilguin 4y ago
  For all i, "Is i a Collatz number?"
- GTP 4y ago
  > I don't think you've shown that Shannon's theorem isn't a problem here so much as the argument presented did not successfully convey that it's a problem. Yes, I was referring to the specific example he made, I wasn't generalizing my conclusion to all possible cases. At the moment of writing my comment I wasn't sure if there was an example that would have worked or not, but the comment about Collatz numbers is a good example. Edit: actually it could be a good example, because wheter that number can be represented in a finite way or not depends on whether Collatz's conjecture is true or false.

by GTP 4y ago

kirkules 4y ago

The presentation in the quote is awkward and doesn't work, as you pointed out, but Lee could instead have just mentioned that there's such a construction with questions of the form "is the answer to the previous question YES [NO]?" which is not constant (not all 1s), and does have the property he wants, i.e. the are such sequences that cannot be specified finitely (corresponding to binary sequences with that property). I don't think you've shown that Shannon's theorem isn't a problem here so much as the argument presented did not successfully convey that it's a problem. Tangentially, I'm interested to see a formal statement/proof of the form of Shannon's theorem add used in this article, because I can sort of intuitively see why it makes sense but I'm not familiar enough with the technical details to guarantee that this is even an appropriate use of the theorem, assuming the yes/no question construction has all the properties Lee wants it to have.

pencilguin 4y ago
For all i, "Is i a Collatz number?"
GTP 4y ago
> I don't think you've shown that Shannon's theorem isn't a problem here so much as the argument presented did not successfully convey that it's a problem. Yes, I was referring to the specific example he made, I wasn't generalizing my conclusion to all possible cases. At the moment of writing my comment I wasn't sure if there was an example that would have worked or not, but the comment about Collatz numbers is a good example. Edit: actually it could be a good example, because wheter that number can be represented in a finite way or not depends on whether Collatz's conjecture is true or false.