The link says not definable in the language of set theory. I take that to mean not definable in principle, not just due to any particular language limitation.
Well, everything is definable in a suitably rich language (you could always just use a language that happens to have a term that refers directly to it)…
But, sure, we could take it as a natural principle that the mathematical objects finitely definable in English are the same as those in the usual formal language of set theory.
But what of it? If Prisoner #1 throws a dart at a dartboard, with 100% probability, it will land at coordinates which are not finitely definable in English. Shall we say that he should not be able to communicate these coordinates to Prisoner #2?
We could, if we want. That’s one way to interpret the problem. But I did intend to discuss another natural interpretation, where he would be able to communicate those coordinates (that infinite quantity of data, so to speak) without difficulty.
Well, outside of the definition “the coordinates where this dart landed”, of course. 
Interpretation of English terms is not a static thing. But there will be no English term which could have communicated those coordinates before the dart was thrown, or to anyone who doesn’t already know where the dart landed.
Let me put it another way: what are we talking about communicating? A certain subset of possible label-sequences. But label-sequences can be thought of as just special real numbers in [0, 1] (e.g., by taking them to be descriptions in base 3, with white being the digit 0 and black being the digit 2). So what we’re talking about communicating is a certain subset of the real interval [0, 1]. Well, this can be encoded by a function from [0, 1] to R which takes some special indicator value on and only on that subset.
So, even if we demand a solution where Prisoner #1 picks a deterministic winning strategy and then communicates it to all the others, he can just say “Are you ready guys? A crazy waveform is about to come out of my mouth, which can be thought of as a function from [0, 1] to R. Pay real close attention to it, and then interpret it as Indistinguishable outlined above.”
[Now, the objection might be raised that this waveform will likely be discontinuous, and thus should be disqualified on physical grounds. To which I can only sigh: yes, it will be discontinuous (proof outside of the scope of this thread). If you restrict prisoners to deterministic winning strategies which can be finitely defined in English augmented with the ability to refer to arbitrary continuous real-valued waveforms of finite duration, well, I can’t help you (save to outline a proof that no such thing can be done if “English” is represented by the usual formal language of set theory). But such restrictions are as little in the spirit of the problem as saying “Surely, at some distance, the far-off labels become indistinguishable to the eye; the prisoners should not be able to act upon this infinite information in its entirety”.]
Well, that’s not the sot of definition that will do the prisoners any good. Maybe “construct” is a better term. I’m going to quote Cabbage from some 6 years ago in a discussion on the continuum hypothesis as an example (my bold):
The bold part is an example of what I’m concerned about. In between “does not exist” and “exists, and here’s how you can in principle find it” is this nebulous “we know it exists, but we can’t actually construct an example.” Not just “don’t know how to”, but “can’t”.
I’m not worried about how he will communicate his choice.
You’re worried about what exactly, then?
Incidentally, I still wonder, how do you respond to the story of the islanders in post #124?
For the sake of having a name, let’s call the objects in question which the axiom of choice ensure us exists “choice-functions” (selecting one sequence from each winning set).
Your worry doesn’t appear to be as to whether choice-functions exist or not; you seem willing to grant that (which, note, is the granting that a winning strategy for this game exists, whether or not our poor prisoners are able to come up with it). And your worry, you’ve just said, isn’t as to how Prisoner #1, were he to know a choice-function, could communicate it to the others.
So I assume your worry is “How could Prisoner #1 come to decide upon a choice-function?”. And you don’t seem to be satisfied by “Well, they’re out there; they exist. Let him just pick one at random”.
But why not? Is this any worse than saying “Let him pick a generic real at random”? [In the sense of “construct” referred to by Cabbage, a generic real is one which cannot be “constructed”; i.e., no closed term in the usual formal language of set theory defines it. With 100% probability, a real chosen at random from [0, 1] is generic (since only countably many reals aren’t).]. Would you object to his ability to do that?
At any rate, are you willing to grant at least “There exist winning strategies, even though I can’t demonstrate to you any explicit deterministic construction of one”? I.e., since you seem unconcerned with the communication aspect and, unless I have missed something, have not rejected the axiom of choice, it would seem you must accept “There is something the prisoners could be told to do which would guarantee them a win”, no?
(In case my tone is off-putting, I want to link to post #139 again; my point isn’t to attack your interpretation, as such, but merely to understand where you are coming from)
I’ve consolidated this post into the above one. Of course, it doesn’t actually do any good, since this reminder is left…
It occurs to me that I’ve given two slightly different definitions of “generic real” in this thread. Well, whatever; either one makes my point.
Yes, I’m also aware that I have this terrible habit of spurting everything out across multiple small posts replying to myself, instead of having the patience to just get it all into one good post. Maybe I’ll make a conscious effort to do that from now on.
I’d answer A, but that doesn’t resolve my concern.
Just what I said in my last post. That knowing the sets exist doesn’t imply that such a set can be constructed, analogously to the case in Cabbage’s post I quoted (but forgot to link to).
In the case of the Islanders above, you already have the set of Islanders, and they each only need pluck one hair. A set of one can be constructed. What you’re asking for in the prisoner problem is much harder. (If the islanders each have hairs with the cardinality of the reals, and need to pluck a number of hairs larger than countable, but less than the cardinality of the reals, that would be a harder problem to solve.)
But if the islanders are able to solve their problem, can’t Prisoner #1 just run in his head a simulation of the islanders to solve his own problem? [Note that the islanders can sometimes only solve their problem by collectively creating a non-"construct"able set; for example, like I said, if the islanders correspond to “winning sets”, with each particular one’s hairs corresponding to the label-sequences within that set, then for each to pluck a single hair is to create the choice-function the prisoners need.]
What are you taking the notion of “constructed” to mean? Unless I am mistaken, Cabbage is using the term to mean “denotable by a closed term in the formal language of set theory”; i.e., definable in a very particular language. But what is the relevance of that concept to whether the prisoners can win or not?
As an analogy, like I said, consider the fact that only countably many reals are "construct"able, and therefore almost all aren’t. What would be the implications for the prisoners if what Prisoner #1 had to come up with was a non-"construct"able real?
I don’t see that that follows.
Specified in some meaningful way that says what the sets are, at least in principle. I don’t see that AC shows that this is possible, even though it shows that those sets exist.
I think we’re just going in circles at this point.
It depends on what “Specified in some meaningful way that says what the sets are” means. If you allow me to use an infinite string of English to specify these infinite objects, it’s easy to do. If you limit me to finite strings, well, then I can’t give a full description.
Just like if I throw a dart a board and you want to know where it landed, I probably can’t give you a full description of its coordinates in a finite string of English. (“Well, it’s approximately <0.23780295, 4.23409>.” “Is that all?” “No, there’s more digits. It keeps going”). It doesn’t mean I couldn’t hit those coordinates; clearly, I just did. It just means I can’t quickly describe them.
We probably are just going in circles at this point. I’ll just explain what I meant by “Prisoner #1 can just run in his head a simulation of the islanders to solve his own problem”:
Prisoner #1 imagines an island with lots of hairs on it, one hair for each possible label-sequence (imagine the sequence is written on the hair). These hairs are then separated into different groups, with hairs put in the same group just in case the label-sequences written on them agree from some point on.
Then Prisoner #1 imagines sticking a different bald person in the middle of each group of hairs, and grafting them onto his head.
Prisoner #1 now thinks “Alright, suppose each of these newly hirsute islanders is asked to pluck a single hair from his head. What might happen?”.
Lots of things might happen, depending on which hairs get plucked. But, presumably, Prisoner #1 is able to simulate in his head the selection of an arbitrary choice of hair by each islander, and thus end up with some selection of plucked hairs.
But the plucked hairs at the end of the simulation will describe just such a set of representative label-sequences as Prisoner #1 needs.
Is there something preventing Prisoner #1 from carrying out such a simulation in order to create, in his head, the set of representative label-sequences he needs?
(Anyway, if you don’t feel like discussing this any further, that’s fine. Might as well just let the thread fade)
OK, I see what you’re saying, and I’m almost there. I think this does specify the set in the way I was asking. I still have a question, though, but I need to specify your example a little more.
I’ll again take the strings to be “zero” to “nine”, so each possible sequence is a decimal number between 0 and 0.99999… We’ll give each islander a Mohawk, so their hairs are all in a line, in order. Inpi’s identifying number is 0.3183098861…, so he has that hair, but he also has the ones where the first digit is different, so he has 0.0183098861…, …, 0.9183098861… Similarly he also has the ones where the first two are different, 0.0083098861…, …, 0.9983098861…, and so forth. So he has a countably infinite set of hairs, all ending the same. These hair numbers are dense over the range 0 to 1.
His friend, Sqrth, has a different identifying number 0.70710678118655… , and also has the ones 0.00710678118655… to 0.90710678118655…, and 0.00710678118655… to 0.99710678118655…, and so forth, again an infinite set all ending the same. His other infinity closest friends all also have a different identifying number, distinct from any of the numbers on anyone else. Taken together, the hairs cover all the numbers 0 to 1 (with only two people overlapping, whom we’ll ignore).
I had been imagining that the chosen hair numbers had to, between them, cover all the possibilities for the leading sequences of digits, and then they wouldn’t be independent. That would be more complicated, but isn’t necessary.
Anyway, Inpi needs to pick one hair, so he decides the first digit is 1, and the second is 2, and then gets bored and just keeps alternating 1 and 2. Sqrth decides to just copy Inpi, so he alternates picking 1 and 2 also. Everyone else is unimaginative, so they just copy Sqrth.
Only one Islander actually has 0.121212…, but they all have numbers infinitesimally close. There’s no finite number of digits after which they need to switch to their ending sequence, so it doesn’t seem that they are limited to a finite number of 1s and 2s.
I need to go to bed, so instead of writing a decent question, I’ll just ask “What’s up with that?”
One warning: the digit sequences 0.09999… and 0.10000… will be different, even though they represent the same number. [We can rectify this by, for example, only using the digits 0 and 9 (i.e., using a Cantor space instead of the entire unit interval), or we can just leave it alone as long as it isn’t bothersome].
The bolded part is, of course, an erroneous inference, at least in the sense that they are forced to use only a finitely long initial string of alternating 1s and 2s.
Inpi isn’t allowed to pick 0.12121212… since it isn’t actually one of his hairs. He can pick a hair that agrees with it up to the 10th digit, or he can pick one which agrees with it up to the millionth digit, or whatever, but he can’t pick one that agrees with it all the way.
It is true that there’s no finite upper bound on how many digits of agreement Inpi is able to get. All the same, he can only get finitely many digits of agreement. (Just like if I asked Inpi to pick a rational number, he could agree with π for an arbitrarily large finite number of digits, but he couldn’t agree all the way. Or, if I asked him to pick a natural number, he could pick an arbitrarily large one, but no matter what, he’d pick something finite.).
It’s understandable that there’s a “What’s up with that?” in there, but it’s just the difference between “for every natural number n, there exists a hair h of Inpi’s such that h agrees for at least n many digits” (which is true) and “there exists a hair h of Inpi’s such that for every natural number n, h agrees for at least n many digits” (which is false). The order of the quantifiers matters; you can’t deduce the latter from the former.
Ah, but you did realize this already; you mentioned “only two people overlapping”. My apologies for missing it.