Bad pun about uncountable infinity
Jul. 21st, 2008 12:50 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
jackQ. OK, so what about uncountable infinities in magic?
What about it?
Q. Does it make infinite combos less degenerate? Does it even make sense?
Well, you may need some tweaks to the rules, because a lot of them are predicted on the idea of things happening one after another. But suppose, eg. you just go round an infinite loop ω1 times.
Q. Well, that was boring.
OK, ok. Suppose you can't do that. Try this. Doubling season says "Whenever you put a counter into play, instead put two of them into play." Suppose you get an infinite number of doubling seasons into play (eg. jumping through a few hoops and an infinite amount of mana to put a copy of it into play an infinite number of times).
Then put a token into play. I think the infinite number of doubling seasons produces an uncountable number of new counters. (Either an uncountable number of new creatures, or almost cooler, one creature with enough +1/+1 counters on that it has power and toughness ℵ1/ℵ1 :))
Think about this either as 2^ω counters, or by counting the counters individually: number the first counter 0, and the counter produced by the first double 0.1 and the counters produced by the second double 0.01 and 0.11, and so on.
Q. Right... Is that rigorous?
I think so... You need either to be comfortable with an infinite stack, (the order things come off it might be a problem?) or to have events happen "in parallel" when it's obvious what's going to happen.
Q. Is it interesting?
I don't know. Maybe. Probably not. It does mean that if you gain uncountably infinite life, your opponent has to jump through some extra hoop, it's not enough just to have an infinite combo. You don't just need a way of gaining infinite cards, infinite mana, and infinite damage, you need an infinite number of copies of some doubling effect as well.
Q. So, specific cards?
Aforementioned doubling season, together with a card that makes it a creature (opalescence) -- or anything else if you can copy it -- and a reusable card that puts a copy of target creature or artifact (kiki-jiki "tap to put into aply a copy of target creature" plus a way to untap it, mirrorweave "all creatures become a copy of target creature" plus infinite creature tokens, spitting image "pay mana and discard a land to repeatedly put a copy of target creature into play").
Isn't there some red enchantment which does twice as much damage?
Q. I ask the questions here.
That's not a question.
Q. I ask the questions here. Right, asshole?
What about it?
Q. Does it make infinite combos less degenerate? Does it even make sense?
Well, you may need some tweaks to the rules, because a lot of them are predicted on the idea of things happening one after another. But suppose, eg. you just go round an infinite loop ω1 times.
Q. Well, that was boring.
OK, ok. Suppose you can't do that. Try this. Doubling season says "Whenever you put a counter into play, instead put two of them into play." Suppose you get an infinite number of doubling seasons into play (eg. jumping through a few hoops and an infinite amount of mana to put a copy of it into play an infinite number of times).
Then put a token into play. I think the infinite number of doubling seasons produces an uncountable number of new counters. (Either an uncountable number of new creatures, or almost cooler, one creature with enough +1/+1 counters on that it has power and toughness ℵ1/ℵ1 :))
Think about this either as 2^ω counters, or by counting the counters individually: number the first counter 0, and the counter produced by the first double 0.1 and the counters produced by the second double 0.01 and 0.11, and so on.
Q. Right... Is that rigorous?
I think so... You need either to be comfortable with an infinite stack, (the order things come off it might be a problem?) or to have events happen "in parallel" when it's obvious what's going to happen.
Q. Is it interesting?
I don't know. Maybe. Probably not. It does mean that if you gain uncountably infinite life, your opponent has to jump through some extra hoop, it's not enough just to have an infinite combo. You don't just need a way of gaining infinite cards, infinite mana, and infinite damage, you need an infinite number of copies of some doubling effect as well.
Q. So, specific cards?
Aforementioned doubling season, together with a card that makes it a creature (opalescence) -- or anything else if you can copy it -- and a reusable card that puts a copy of target creature or artifact (kiki-jiki "tap to put into aply a copy of target creature" plus a way to untap it, mirrorweave "all creatures become a copy of target creature" plus infinite creature tokens, spitting image "pay mana and discard a land to repeatedly put a copy of target creature into play").
Isn't there some red enchantment which does twice as much damage?
Q. I ask the questions here.
That's not a question.
Q. I ask the questions here. Right, asshole?
no subject
Date: 2008-07-22 03:34 pm (UTC)I don't think your proposed proof by induction works. I think the inductive step is fine, but your initial step is somewhat missing. (And in fact undefined, since you're trying to induce downwards from infinity to zero, which doesn't work too well.)
I've got myself a little confused about this, since I can't see why the Dismantle example above can produce 2^ℵ0 if the set of ℵ0 Doubling Seasons doesn't. You do seem to have provided an injection from your infinitely-Doubled tokens into the rationals, but equally it seems to me that "take 1 and double it ℵ0 times" is what you're doing, and surely that must produce 2^ℵ0 tokens?
no subject
Date: 2008-07-22 04:04 pm (UTC)Incidentally, I'm not at all sure what you mean by this. You're switching between ordinals and cardinals rather carelessly and this may be what's confusing you.
In particular, taking a number and doubling it repeatedly is a series of operations that have to happen in some order. So you must count the doublings with an ordinal, not a cardinal.
The order matters because you might get a different result if you do the doublings in a different order. For example, even though |ω| = |ω+1| = ℵ0, we have 2ω = ω, but 2ω+1 = 2ω.