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Earlier in the game an Infinity Elemental was given Lifelink and dealt damage, giving me infinite life. I have a Form of the Dragon in play, so that at my end step my life total becomes 5.

Rule 119.5 states

If an effect sets a player’s life total to a specific number, the player gains or loses the necessary amount of life to end up with the new total.

The ruling on Infinity Elemental states

If an Infinity Elemental you control gains lifelink and deals damage, you’ll then have infinite life. From that point on, your life total effectively can’t change. If you lose life, you’ll still be at infinite life. You can pay any amount of life and still be at infinite life. In fact, you can be hit by an opposing Infinity Elemental and still be at infinite life.

On one hand, "the necessary amount of life to end up with the new total" seems to indicate that I must end up at 5 life, but the ruling seems to indicate that not even an infinite loss of life can change my life total (infinity minus infinity being mathematically undefined aside). Which one controls?

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There is a ruling on this already in gatherer. You said it was rule 119.5 but it is also a ruling on this card. So in this situation your life becomes 5 at the end of each turn regardless of being set to infinity because of life link.

If an effect sets your life total to a specific number, it will be that number, even if you previously had infinite life.

The other gather ruling even makes reference to the ruling that is right below it which I have listed above.

If you give Infinity Elemental lifelink and it deals damage, you will gain infinite life. You might think infinite life sounds like a lot, and you’d be right. Once at infinite life, your life total mostly can’t change (see below). At infinite life and gain 2 life? You’re at infinite life. At infinite life and lose 29 life? You’re at infinite life. You can pay any amount of life and still be at infinite life. In fact, if you’re at infinite life and get with an opposing Infinity Elemental, you’d still be at infinite life.

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    Silly me didn't read the Gatherer rulings; I was looking at Scryfall's entry, which doesn't have that ruling.
    – Purple P
    Commented Jan 30, 2021 at 22:50
  • Wouldn't this also mean that you'd lose infinite life when this occurs, potentially creating an infinite number of "whenever a player loses life" triggers?
    – nick012000
    Commented Jan 31, 2021 at 10:02
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    @nick012000 No; "whenever a player loses life" refers to any amount of life loss happening in one instance. Losing 10 life as a result of being hit by a 10-power creature would just trigger your "whenever a player loses life" once, not ten times.
    – Patrick Stevens
    Commented Jan 31, 2021 at 14:19
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IMHO (we can't be sure; it's an Un- card after all), the rulings on Infinity Elemental reflect the 'common' properties of the mathematical concept of infinity (Aleph null), at least partially: infinity plus or minus a finite number is still infinity. The fact that "In fact, you can be hit by an opposing Infinity Elemental and still be at infinite life." surprises me; infinity minus infinity is usually undefined, not infinity.

It's safe to say the comprehensive rules don't have to say anything about this, but if I would need to house-rule this: I would say that Form of the Dragon trumps Infinity Elemental, since it doesn't modify your life total by adding or subtracting, but rather by setting it to a specific number.

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  • "rather by setting it to a specific number" - Now that I think about it, rule 119.5 basically says setting is the same thing as adding or subtracting. Maybe both the rule and the ruling say the same thing.
    – Purple P
    Commented Jan 30, 2021 at 22:45
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    @PurpleP In this context I read 119.5 as "Setting your life total will trigger triggered abilities and be modified by replacement effects that care about adding to or subtracting from your life total, but it is not done through addition or subtraction; the life total is simply set."
    – Arthur
    Commented Jan 31, 2021 at 8:23
  • It would probably be possible to make these subtractions well-defined by taking the infinity to mean ω, the smallest infinite ordinal, rather than ℵ_0 (Aleph null), the smallest infinite cardinal. Because the ordinals are defined in such a way that a < b is well-defined for all ordinals a, b (even infinite ones), we can represent numbers like 2ω+3, whereas 2 * ℵ_0 is not meaningful. There would still be some complications (since addition of infinite ordinals isn't commutative), but it's a step in the right direction.
    – Ray
    Commented Jan 31, 2021 at 18:56
  • @Ray 2 * ℵ_0 is entirely meaningful. The result is just ℵ_0. Cardinal numbers have an arithmetic to them, with addition, multiplication and exponentiation, just like ordinal numbers. But for infinite cardinals (at least assuming the axiom of choice), addition and multiplication is rather boring, and the result is just the bigger one of them. If you want subtraction to work well with infinities, my personal suggestion is to use the surreal numbers. Just because they are so darned fun.
    – Arthur
    Commented Jan 31, 2021 at 19:07
  • @Arthur Quite true. I originally had an explanation of why that was, but couldn't fit it in 600 characters. But with 600 more: The cardinal numbers represent the number of elements in a set. Two sets have the same number of elements iff we can establish a one-to-one relation between them: since there are ℵ_0 natural numbers, any set where we can uniquely label the elements as 0, 1, 2, ... has size ℵ_0. If A and B are both ℵ_0, we can number A union B as: 2i+1 -> B_i, 2i -> A_i, so A union B is size ℵ_0. The key difference with the ordinals is that ω-ω is well-defined, but ℵ_0 - ℵ_0 isn't.
    – Ray
    Commented Jan 31, 2021 at 19:21
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Given that there will be no official answer for rule questions about silver-bordered cards; in my opinion you should have 5 life.

It is unclear whether 119.5 is describing the mechanism by which you follow the instructions to set your life to a specific value, or whether it I’d describing a natural consequence that comes from setting your life to a specific value.

The latter is the more natural interpretation to me, and if interpreted this way, then there is no contradiction. You set your life total to 5, and 119.5 says that this counts as losing as much life as necessary for that to happen (which would be an undefined or infinite amount). It doesn’t matter whether that amount is a real number or not; 119.5 is just explaining that it counts as loss of life.

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