Swammerdami
Squadron Leader
Most readers of this subforum are familiar with the Axiom of Choice, for which a minimal statement might be
There is a famous puzzle about 100 prisoners each trying to guess his own hat color. Prisoner #19, for example, can see the hat colors of #1, #2, ... #18, and can hear the guesses of prisoners #100, #99, ..., #20 as they try to survive.
The puzzle has nothing to do with the Axiom of Choice but try to solve it before reading on. Acting together, the prisoners can devise a plan such that at least 99 of them will live. (Best is to assume that if 99 or more guess correctly, then they ALL live — with that rule change we need not depend on altruism.)
Now let's make the problem much more difficult. Instead of two hat colors, there will be 1000 colors. (We could make it countably infinite instead.) Instead of just 100 prisoners, there will be infinitely many: a prisoner #N for each natural number N. AND each prisoner will NOT be able to hear the guesses or gunshots before his guess: When it is one prisoner's turn to guess he will have zero information to go on except the (usually infinitely many) hat colors he can see, along with any policy he pre-agreed with his mates. He has only 1 chance in 1000 to survive, right? For every billion prisoners, about 999 million will die, right? Wrong! IF you accept the Axiom of Choice.
Postulating the Axiom of Choice, can the prisoners devise a policy such that at most a finite number of them misguess? Most of the prisoners will be looking at an infinite number of hats. Play along please; treat this as a thought experiment where each prisoner has a magic supercomputer that copes with infinite and even uncountable sets.
The answer is, Yes! — an answer so mind-boggling that it might make you assume the Axiom of Choice is clearly false! I will post the very simple solution in a few days if nobody beats me to it.
There are other paradoxical conclusions that can be derived from the Axiom of Choice.
(Note that the Axiom of Choice is never needed for FINITE sets. If I write "If there are articulable mythologism models, THEN you should be able to choose one to actually, well, articulate" I am NOT dependent on the Axiom of Choice as long as I insist that the articulation of the model require no more words than are in all the volumes of the Library of Congress.)
∅∉S ⇒ ∃f : ∀c∈S, f(c)∈c
There is a famous puzzle about 100 prisoners each trying to guess his own hat color. Prisoner #19, for example, can see the hat colors of #1, #2, ... #18, and can hear the guesses of prisoners #100, #99, ..., #20 as they try to survive.
The puzzle has nothing to do with the Axiom of Choice but try to solve it before reading on. Acting together, the prisoners can devise a plan such that at least 99 of them will live. (Best is to assume that if 99 or more guess correctly, then they ALL live — with that rule change we need not depend on altruism.)
Now let's make the problem much more difficult. Instead of two hat colors, there will be 1000 colors. (We could make it countably infinite instead.) Instead of just 100 prisoners, there will be infinitely many: a prisoner #N for each natural number N. AND each prisoner will NOT be able to hear the guesses or gunshots before his guess: When it is one prisoner's turn to guess he will have zero information to go on except the (usually infinitely many) hat colors he can see, along with any policy he pre-agreed with his mates. He has only 1 chance in 1000 to survive, right? For every billion prisoners, about 999 million will die, right? Wrong! IF you accept the Axiom of Choice.
Postulating the Axiom of Choice, can the prisoners devise a policy such that at most a finite number of them misguess? Most of the prisoners will be looking at an infinite number of hats. Play along please; treat this as a thought experiment where each prisoner has a magic supercomputer that copes with infinite and even uncountable sets.
The answer is, Yes! — an answer so mind-boggling that it might make you assume the Axiom of Choice is clearly false! I will post the very simple solution in a few days if nobody beats me to it.
There are other paradoxical conclusions that can be derived from the Axiom of Choice.
(Note that the Axiom of Choice is never needed for FINITE sets. If I write "If there are articulable mythologism models, THEN you should be able to choose one to actually, well, articulate" I am NOT dependent on the Axiom of Choice as long as I insist that the articulation of the model require no more words than are in all the volumes of the Library of Congress.)