A game strategy that depends on the Continuum Hypothesis?
Thanks to @jeanas on Mathstodon, here’s a fun pair of problems, the first easy, the second not so much.
Suppose Sam and Joe are playing a game where Sam is given two finite ordinals, and has to give one to Joe. Joe then is allowed to make finitely-many guesses at the other one, and they both win if a correct guess is made. One can find an always-winning strategy for this game.
Now suppose Sam and Joe are given three ordinals, not necessarily finite. Sam has to give Joe two of them, and Joe has to make finitely-many guesses at the third. Otherwise the game is the same The challenge (as I understand it) is now to show that there is a winning strategy if and only if all three ordinals are countable.
However, the way I was originally told about this game was in terms of picking real numbers, not ordinals, and that the answer was dependent on the Continuum Hypothesis being true or not—which is famously not provable nor refutable from the ZFC axioms of set theory. The hint was that one should use the well-ordering principle, which immediately clued me in that the phrasing in terms of real numbers was a red herring. Using the Axiom of Choice, then if (and only if) the CH is true, then there is a bijection , where is the set of all countable ordinals. Since the solution to the first problem really uses the well-orderedness of the set of all finite ordinals (namely, there are no infinitely-long, strictly descending sequences of ordinals, finite or otherwise), this is a big hint for the second. Otherwise, just stating the problem as being about real numbers hides the solution behind both AC and CH, and there are no useful order properties of the reals with their usual ordering here. It feels like a gotcha to state this in terms of real numbers, when thinking up the actual winning strategy is more fun once you have the non-obvious secret step revealed, since all the information is now available. It’s not the strategy that’s independent of ZFC, but rather the translation trick that requires a strong axiom that’s also independent from the other axioms that make up the ZF fragment.
#game #ordinals