After yesterday’s exertions on the golf course, I took it a bit easier today. Mostly I worked on Darths & Droids story planning, but I took a lunch break to walk up to the local shops and get a chicken burger for lunch.
I also had some interesting discussions with friends in our online chat. Some of it was Darths & Droids story planning, so I won’t go into that further. But somehow we segued into a discussion of the phasing rules in Magic: the Gathering – I think prompted by Mark Rosewater’s latest design article, in which he says:
We’re experimenting with making phasing deciduous.
Okay, this probably makes no sense if you don’t know the early history of Magic: the Gathering, but bear with me. Phasing is a rule that first appeared in the game in 1996, but which was considered too confusing and cumbersome to use again. But now they’re playing with bringing it back, at least in a limited way. (“Deciduous” in the above quote means a rule mechanic that they always consider available to include in new card sets if it makes sense for that set.)
Phasing, in essence, is an effect that makes cards in play behave as though they are not in play – they “phase out” for a turn and then reappear. While phased out, nothing can affect them, nor can the phased out card affect anything else. It’s as if they are briefly shunted to another reality.
In the ensuing discussion, I said they shouldn’t merely have one “alternate reality” – things should be able to phase into specific other realities, of which there could be several… or even infinitely many. Then if you have two infinite sets of alternate realities orthogonal to one another, and you reference them by real numbers (i.e. all the integers, rationals, algebraic irrationals, and transcendental numbers), you could phase all of your creatures in such a way that you could duplicate them using the Banach-Tarski theorem. (For a reminder on why that premise leads to that conclusion, refer to my Irregular Webcomic! annotation on the Banach-Tarski theorem.)
Someone of course immediately pointed out that you can only use the Banach-Tarski theorem if you assume the axiom of choice to be true. (For a simple primer on the axiom of choice, see my annotation on that.)
Then someone else said that rule 722.2a of the Comprehensive Rules of Magic: the Gathering (June 1, 2020 edition) might actually imply the axiom of choice. Rules 722.2a says:
722.2a At any point in the game, the player with priority may suggest a shortcut by describing a sequence of game choices, for all players, that may be legally taken based on the current game state and the predictable results of the sequence of choices. This sequence may be a non-repetitive series of choices, a loop that repeats a specified number of times, multiple loops, or nested loops, and may even cross multiple turns. It can’t include conditional actions, where the outcome of a game event determines the next action a player takes. The ending point of this sequence must be a place where a player has priority, though it need not be the player proposing the shortcut.
Example: A player controls a creature enchanted by Presence of Gond, which grants the creature the ability “{T}: Create a 1/1 green Elf Warrior creature token,” and another player controls Intruder Alarm, which reads, in part, “Whenever a creature enters the battlefield, untap all creatures.” When the player has priority, they may suggest “I’ll create a million tokens,” indicating the sequence of activating the creature’s ability, all players passing priority, letting the creature’s ability resolve and create a token (which causes Intruder Alarm’s ability to trigger), Intruder Alarm’s controller putting that triggered ability on the stack, all players passing priority, Intruder Alarm’s triggered ability resolving, all players passing priority until the player proposing the shortcut has priority, and repeating that sequence 999,999 more times, ending just after the last token-creating ability resolves.
The argument is that it is not only possible within the rules of MtG to produce a loop of actions, but nested loops of actions, and at each loop this rule says you can specify how many times the loop is executed. If the nest of loops is infinitely deep, this means that you are effectively choosing an element from each of an infinite number of sets, where each set contains an infinite number of elements. The rules of the game say you can do this. Therefore the rules of the game say that you can apply the axiom of choice.
This is, in mathematical terms, a rather simplistic case and doesn’t (I believe) in fact rely on the axiom of choice to be doable in an actual game (although I may be wrong), but that didn’t stop us having a fun discussion about it. It was topped off by the original proposer of the example of rule 722.2a saying:
I’m not sure what it says about us that I can say “the Magic: the Gathering comprehensive rules imply the axiom of choice” as a throwaway joke, and the responses are “your rule numbering is out of date”, “no they don’t” and “actually maybe they do” (and not, for example, “ha”, “what the fuck”, or “you nerd”).
This is nowhere near the nerdiest argument we’ve ever had, by the way…
New content today:
Hmm. If I understand your strategy correctly, you’re defining, for each x, loop x to read:
“Run loop x+1 once, then choose an element of set x.”
That will only give you the axiom of countable choice, which is rather weaker than true AC.
Also, the rules say that “The ending point of this sequence must be a place where a player has priority”. This might cause you trouble in the way you’ve set things up, as it requires the sequence of plays to have an end point – so it can’t just crash or be undefined how to play it. And unfortunately, it’s problematic to do that when your collection of subloops is illfounded, like the one I outlined above.
Essentially, you’re going to start out following it and get stuck being sent down and down into deeper and deeper subloops for an infinite amount of time. Spending an infinite amount of time running something isn’t a problem in itself in set theory, but once you’ve spent infinitely long doing that, there’s no obvious instruction to follow next. If I’m going to obey the instructions in all the loops, I’m going to have to obey start at the “end” of the infinite sequence and work my way outwards, which is equivalent to starting at the top of the natural numbers and counting down to zero without missing any numbers out, which is impossible. I think the most conventional interpretation is that the program will just crash instead.
We could try to solve this by swapping around the order of the instructions, so we start by picking the element of set x and then run loop x+1. But then the program would start by picking out an element of each set, then crash, and not have an endpoint where someone has priority. You could solve it by saying something like “if you start opening an infinite chain of subloops then once they’re all open just close them all instantly” but…eh, that’s kind of messy.
I think a more conventional approach would be to forget the nested subloops, and instead just take the single loop:
“Pick an element of set x, then turn x into x+1.”
and run it omega many times. (omega is set theorist speak for the smallest infinite number, which also happens to be the set of natural numbers.) The rules, after all, just say the loop has to repeat a specified number of times, not that that number has to be finite. If you have some sort of convention for what to do with your variable x when you reach a limit ordinal, then you could get the axiom of choice for any ordinal number of sets, rather than just countable choice. I can’t remember off the top of my head whether that’s equivalent to full AC or still a bit weaker.
You might also have some trouble with the wording at the start, you have to “describe” some sequence of game choices. Is it sufficient to just say “pick one arbitrarily”?
Speaking of ambiguous wordings, we could also interpret “The ending point of this sequence must be a place where a player has priority” as being a statement of fact rather than a condition the sequence has to satisfy before we can use rule 772.2a. As in “The sides of this right angled triangle must satisfy a^2+b^2=c^2.” If so, then depending on what you can do in Magic (I haven’t played it) we would probably be able to prove the axiom of determinacy! Find some situation where a player gets priority at the end of an infinite sequence of Magic moves if and only if their moves represent a winning strategy in your game.