Uncountable infinity in magic
Jul. 23rd, 2008 02:27 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
jackI've wracked my brains a bit on the subject, and I left it at about this point:
* Repeated doubling one after the other can never produce an uncountable infinity: you can always count these since there are a finite (or countable) number produced at each stage
* Repeated doubling backwards, as using the stack, doesn't seem well-defined. If at each (backwards) time t, there are twice as many counters as at t+1, then you could say that a set of counters numbered 0.(0..0t+1)x become counters numbered 0.(0..0t)x, but I can't see any way of specifying what the set of potential x is, the definition is consistent with any set defined by the tails of x.
* It's not in principle impossible. There could in theory have been a card that says "do x ... for each legal target", which for a spell which can target any number of creatures, with an infinite number of creatures, would target every possible subset. That card would not play well, it would be crazy even for finite numbers, but it would be eminently possible.
* I can't think of any way to achieve an uncountable. I considered and discarded a few ideas:
* Ink-Treader Nephilim says "Whenever a spell ... only target ...copy the spell for each other creature that spell could target." The rulings say that if a spell has multiple targets, but they all target Nephilim, it is still copied. However, it is still only copied once for each other creature, not for each legal target.
* Grip of Chaos says "Whenever a spell ... has a single target ... reselect its target at random ..." If the same rulings as nephilim apply, you could target it with a spell with any number of targets. It looks like the target would be reselected from all single targets, not all combinations. However, either way, it still says "reselect" not "copy for each".
* If you do infinite doubling, if there were anything which happened once for each sequence of counters put into play, it would count uncountable sequences. I can't think of anything though.
* S.N.O.T (from the non-serious Unhinged/Unglued block) says "When S.N.O.T comes into play, you may stick it onto another creature called S.N.O.T. If you do, they are considered a single creature." If it had been phrased slightly differently, you might have been able to choose whether any collection of S.N.O.T. cards were considered a single creature, in which case any subset of an infinite number of S.N.O.T would form a different creature. However, it doesn't say that.
* Another Unhinged card, "Look At Me, I'm R&D" says to choose two numbers, one one higher than the other, and all instances of the first one are instead the second on. I'm not sure what this would do, but it it feels like it might do something.
* And just for completeness, the other Unhinged card which is always used for dodgy combos Magic:TG was not meant to encompass, "R&D's secret lair", says to ignore all errata, thus allowing you to play any card which was printed with a typo, allowing it to do some insane thing by accident.
* Or, any Unhinged card which lets you bring something in from outside the game, acquiring some characteristic equal to your age, or height, or the cost of a drink you desire, etc; if you're considering a short list of abusable cards, these often let you do something degenerate if you gloss over some small part of the rules :)
* Repeated doubling one after the other can never produce an uncountable infinity: you can always count these since there are a finite (or countable) number produced at each stage
* Repeated doubling backwards, as using the stack, doesn't seem well-defined. If at each (backwards) time t, there are twice as many counters as at t+1, then you could say that a set of counters numbered 0.(0..0t+1)x become counters numbered 0.(0..0t)x, but I can't see any way of specifying what the set of potential x is, the definition is consistent with any set defined by the tails of x.
* It's not in principle impossible. There could in theory have been a card that says "do x ... for each legal target", which for a spell which can target any number of creatures, with an infinite number of creatures, would target every possible subset. That card would not play well, it would be crazy even for finite numbers, but it would be eminently possible.
* I can't think of any way to achieve an uncountable. I considered and discarded a few ideas:
* Ink-Treader Nephilim says "Whenever a spell ... only target ...copy the spell for each other creature that spell could target." The rulings say that if a spell has multiple targets, but they all target Nephilim, it is still copied. However, it is still only copied once for each other creature, not for each legal target.
* Grip of Chaos says "Whenever a spell ... has a single target ... reselect its target at random ..." If the same rulings as nephilim apply, you could target it with a spell with any number of targets. It looks like the target would be reselected from all single targets, not all combinations. However, either way, it still says "reselect" not "copy for each".
* If you do infinite doubling, if there were anything which happened once for each sequence of counters put into play, it would count uncountable sequences. I can't think of anything though.
* S.N.O.T (from the non-serious Unhinged/Unglued block) says "When S.N.O.T comes into play, you may stick it onto another creature called S.N.O.T. If you do, they are considered a single creature." If it had been phrased slightly differently, you might have been able to choose whether any collection of S.N.O.T. cards were considered a single creature, in which case any subset of an infinite number of S.N.O.T would form a different creature. However, it doesn't say that.
* Another Unhinged card, "Look At Me, I'm R&D" says to choose two numbers, one one higher than the other, and all instances of the first one are instead the second on. I'm not sure what this would do, but it it feels like it might do something.
* And just for completeness, the other Unhinged card which is always used for dodgy combos Magic:TG was not meant to encompass, "R&D's secret lair", says to ignore all errata, thus allowing you to play any card which was printed with a typo, allowing it to do some insane thing by accident.
* Or, any Unhinged card which lets you bring something in from outside the game, acquiring some characteristic equal to your age, or height, or the cost of a drink you desire, etc; if you're considering a short list of abusable cards, these often let you do something degenerate if you gloss over some small part of the rules :)
no subject
Date: 2008-07-23 09:39 pm (UTC)But am I wrong that, eg. "put a counter into play for each (possibly infinite) subset of the integers" will be uncountable? (As in point 3.) If so, there doesn't seem to be a magic card which says to do that, but that seems to be just a coincidence, not inherently impossible.
(This stuff is tricky to get right. I don't think I've managed one completely correct comment on this topic so far...)
LOL. Yep. But thank you, better than me :)
no subject
Date: 2008-07-23 10:17 pm (UTC)No, you're right.