### Infinity

There is no such thing as infinity in Magic.

I know, this seems a bit unfair, since infinity is such a cool concept, but sadly the rules specifically state, anytime an infinite number of things happen, you just choose a number for how many times they happen, and then it ends. Unless you can’t, then you break the game as badly as if you had just divided by zero. This also is in the rules, and is usually referenced by the old resolving-a-O-ring-with-only-two-other-O-rings-on-the-board example.

What this means is, assuming you and your opponent both have a way of producing infinite mana, he or she will never be able to cast a Fireball big enough that you can’t save yourself with an Alabaster Potion.

But what if we got rid of the rules on infinities? What if you really could produce an infinite amount of mana, not merely an arbitrarily large amount?

Well, first of all we have to make sure we actually can make an infinite amount of mana. Most “infinite mana combos” involve taking a finite number of steps that end up where you started from but with some number of extra mana in your mana pool. If you wanted infinite mana, you’d have to perform these steps an infinite amount of times, which isn’t actually possible. Luckily, the magic rules state that if you can demonstrate a loop like this, it’s enough to point out the sequence and, assuming your opponent can’t or doesn’t want to do anything to interrupt it, name a number of times you want to perform it. So let’s make things easy on us and just say that we modify that rule to allow for that number to be “infinite”.

So, back to where we were. Say we have an infinite amount of mana, and our opponent casts an infinitely large Fireball. Can we save ourselves with an Alabaster Potion?

Well, this is where things get tricky. Infinity is strange in that regular math doesn’t actually work on it. You might think, sure, if you can prevent an infinite amount of damage, then you’re safe, but it doesn’t actually work like that.

Take the set of integers (0,1,2,…), for example. There are an infinite amount of numbers. Now subtract all the even numbers from that set, and you’re left with 1,3,5,…., which is still an infinite amount of numbers. Infinity minus infinity is not defined. So that’s our first problem, and it exemplifies one of the main reasons Magic doesn’t allow for infinities.

But let’s say that we redefine “infinity” in a way that we actually can resolve such a situation. Say that we state that, whenever we create an infinite amount of something, we have to define a set of infinite integers and use that to mark that particular infinity. so, in the example above, the Fireball player defines his or her infinity as the set of integers. Then all the Alabaster Potion player needs to do is define the same set for his or her infinity, and they’l live.

Let’s take another example. Say we still have infinite mana, and a way to recur Gelatinous Genesis. We cast it once for X=1, creating 1 1/1 Ooze. Then we cast it for 2, then for three, and so on, an infinite number of times (again, modifying the rules so that we can do this without actually taking an infinite amount of time). Your opponent has a Deepfire Elemental. Can your opponent kill all your oozes?

Well, the answer to this is… maybe. The problem is, for each value of X you have to choose between one of X creatures to target, which doesn’t sound that bad, until you realize that you, as a person, have to choose which one of those creatures to target. An infinite number of times. And there’s no way to shortcut that, at least not in the Magic rules. Luckily, math comes to the rescue here… sort of.

There is a principle called the Axiom of Choice, which basically says that if you have to make an infinite number of decisions for something, like choosing an ooze, you can. The funny thing is, math doesn’t need the Axiom of Choice. (Most) math works just fine without it, though some proofs get much trickier, which is why a lot of times in some fields of mathematics, you’ll see the phrase “Assuming AC,…” simply because without that phrase the proof just plain doesn’t work, though that doesn’t actually mean that the result isn’t any less true without the AC.

So, let’s add the Axiom of Choice to the comprehensive rules. Now you can kill all the oozes, but will that save you? Maybe, maybe not, because the Ooze player can just cast it again, and you can kill them, and they can cast them, and you can kill them, and so on, ad infinitum. There’s no easy way to resolve this situation, it doesn’t fit the unbreakable-loop rule, so the game goes on forever or until one player gets tired and falls asleep. Then the other player gets the last word in, and wins the little face-off.

…and that’s why we don’t allow “infinities” in Magic.

I’ll have more to say on the subject for sure, but I hope you enjoyed this little excursion.

Exercises:

1. Say instead of Deepfire Elemental above, you have a way to recur Pernicious Deed. Your opponent has cast his or her infinite number of Oozes and passed the turn to you. Can you kill them all? Do you need AC to do so?
2. What other situations can you think of where infinity breaks the game?