"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 qu

"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.