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No, your argument is wrong. The infinite number of prisoners find that it is impossible to express the sequence of all of the hat colors in their native langua

No, your argument is wrong. The infinite number of prisoners find that it is impossible to express the sequence of all of the hat colors in their native language in a finite amount of time, which is a fundamental limit of formal languages. The language lacks the forms to express the state of reality. "Next, the prisoners invoke the Axiom of Choice to pick an element in each equivalence class, which they all agree on and memorize." Not possible. AC is invoked when you can't write down a rule, the purpose of the rule being to "consistently make an infinite number of choices in an unambiguous way, in a representation format that is equivalent to a Turing machine". So if you invoke AC, then you have to give something up, but you haven't given anything up. You're pretending you're in the representable, computable world. Any prior choice of representation system can be thwarted by the actual reality. The information contained in the infinite sequence of hat colors can be so great that representations of it simply don't exist. Proof follows. Let R be the set of real numbers. The equivalence classes described in the article can be extended to binary representations of real numbers, but this doesn't concern us and is a distraction. In this way, each real number would correspond to a different configuration of hat colors according to the values of the binary digits. If a number (like 1) has two binary representations, "0.1111..." and "1.0", then we pick the one that ends after finitely many places (1.0). All we care about is whether certain real numbers exist. Let F be a formal proof system that corresponds with "your intuitive notion of a legitimate existence proof" (Here we take F to be the system of representation the prisoners are using to communicate, so the must all agree beforehand that F is the test that will be used to determine whether or not /a supposed existence and uniqueness proof of a particular real number/ is valid. These are prisoners after all: they might try to convince each other that certain things exist when they really don't exist). Because R is uncountable, and because there can only be countably many valid proofs in F, there exist elements of R whose existence and uniqueness cannot be proven by F. Therefore F "is wrong" and the prisoners lack the ability to express the "state of the world" in which they find themselves.

hyperpape 12y ago
You are importing misleading assumptions about what a strategy is, in this sort of mathematical game (basically, you're thinking of a strategy as a thing that concrete humans beings communicate to each other, rather than as a sort of mathematical object). Of course, given that fact, I don't actually care about this particular thought experiment. Who cares that this abstract object exists? The hats and such are just a way of dramatizing the actual claim about the existence of a particular abstract object.
- dbpokorny 12y ago
  "Who cares that this abstract object exists?" The form of this question is "Who cares about X?" so I would put this back to you, "who cares about answers to questions of the form, 'who cares about X?'?" "The hats and such are just a way of dramatizing the actual claim about the existence of a particular abstract object." The hats are about binary representations, and calling out the hat colors is about being able to compute binary representations. I don't know how to interpret the word "strategy" without implying "algorithm". The setup is that an interrogator sequentially queries the prisoners, and the prisoners have to come up with answers. If you're criticizing the idea that computability should be injected into talk of what a "strategy" is, then we have to get to the heart of the problem: are the prisoners different people or are they all "manifestations of the same underlying strategy"? (I'm saying that the ability to make a prior agreement about an infinite amount of information is tantamount to the prisoners losing their individuality and being "merged" into a kind of consciousness that speaks for all of the prisoners simultaneously). If they are different people, then we have to be able to formalize somehow the concept that information available to one is not available to another. So I turn the question around: how do you propose to model the fact that the prisoners do not have insight into the "raw visual sense data" of the other prisoners, yet the "raw auditory sense data" is shared? Now, back to the original problem: the hats are an attempt to hide under the rug the very difficult concept of being able to evaluate/compute the binary digits of a particular real number given some mathematical description of it. The key word is "agree". The prisoners have to "agree" on something, and in this context, we aren't sure what that means, but I'm taking it to mean that there is a finite alphabet they're using to communicate, they can only communicate finite sentences, and other similar practical assumptions... Since there are uncountably many real numbers and only countably many computable numbers, there are (in ZFC) plenty of real numbers whose existence and uniqueness have been proven, but which are not computable. This is going to turn out to be an insurmountable operational obstacle for the prisoners. For example any Chaitin constant will do. I've added a comment to the original blog post explaining the details.
  - hyperpape 12y ago
    Neither computability nor sense data have any relevance. The prisoners are a literary device. As far as formalizing the lack of information, the elements of each equivalence class chosen need not depend on the particular arrangement of the prisoners--that is more than enough. Again, you're thinking of this far too literally.

hyperpape 12y ago

You are importing misleading assumptions about what a strategy is, in this sort of mathematical game (basically, you're thinking of a strategy as a thing that concrete humans beings communicate to each other, rather than as a sort of mathematical object). Of course, given that fact, I don't actually care about this particular thought experiment. Who cares that this abstract object exists? The hats and such are just a way of dramatizing the actual claim about the existence of a particular abstract object.

dbpokorny 12y ago
"Who cares that this abstract object exists?" The form of this question is "Who cares about X?" so I would put this back to you, "who cares about answers to questions of the form, 'who cares about X?'?" "The hats and such are just a way of dramatizing the actual claim about the existence of a particular abstract object." The hats are about binary representations, and calling out the hat colors is about being able to compute binary representations. I don't know how to interpret the word "strategy" without implying "algorithm". The setup is that an interrogator sequentially queries the prisoners, and the prisoners have to come up with answers. If you're criticizing the idea that computability should be injected into talk of what a "strategy" is, then we have to get to the heart of the problem: are the prisoners different people or are they all "manifestations of the same underlying strategy"? (I'm saying that the ability to make a prior agreement about an infinite amount of information is tantamount to the prisoners losing their individuality and being "merged" into a kind of consciousness that speaks for all of the prisoners simultaneously). If they are different people, then we have to be able to formalize somehow the concept that information available to one is not available to another. So I turn the question around: how do you propose to model the fact that the prisoners do not have insight into the "raw visual sense data" of the other prisoners, yet the "raw auditory sense data" is shared? Now, back to the original problem: the hats are an attempt to hide under the rug the very difficult concept of being able to evaluate/compute the binary digits of a particular real number given some mathematical description of it. The key word is "agree". The prisoners have to "agree" on something, and in this context, we aren't sure what that means, but I'm taking it to mean that there is a finite alphabet they're using to communicate, they can only communicate finite sentences, and other similar practical assumptions... Since there are uncountably many real numbers and only countably many computable numbers, there are (in ZFC) plenty of real numbers whose existence and uniqueness have been proven, but which are not computable. This is going to turn out to be an insurmountable operational obstacle for the prisoners. For example any Chaitin constant will do. I've added a comment to the original blog post explaining the details.
- hyperpape 12y ago
  Neither computability nor sense data have any relevance. The prisoners are a literary device. As far as formalizing the lack of information, the elements of each equivalence class chosen need not depend on the particular arrangement of the prisoners--that is more than enough. Again, you're thinking of this far too literally.