Uncountable infinity in magic

[jack]

I'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..0^t+1)x become counters numbered 0.(0..0^t)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 :)

Comments

[angoel.livejournal.com]

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.

If you do it an infinite number of times it does, and I thought you had mechanisms for doing things an infinite number of times. If you don't have mechanisms for doing things an infinite number of times then I'm not sure how you plan to get any infinite amounts, countable or otherwise.

I'm confused.

[simont]

Perhaps I'm misunderstanding you here (I don't know anything about Magic), but if you take a finite set, repeatedly double its size an infinite number of times, and take the disjoint union of all of the resulting sets, then it's definitely still countable. Proof given in a comment on a previous post in this series.

[cartesiandaemon.livejournal.com]

That's what I meant.

[angoel.livejournal.com]

Yes, I was confused. I was conflating the (countable) rationals with the (uncountable) reals. Silly boy.

[gareth-rees.livejournal.com]

I think that what you're trying to do is impossible: starting only with ω, you can't get an uncountable collection of items by any countable sequence of ordinal additions, multiplications or exponentiations. More to the point, any ordinal which can be generated by a set of explicit rules over these operations is known as a recursive ordinal, and all recursive ordinals are smaller than the first Church–Kleene ordinal, which is countable. (Wikipedia has a reasonably clear page on large countable ordinals which explains this.)

If you want to get an uncountable number of things then you are going to need to go beyond basic algebraic operations and invent some kind of completeness operation (like a supremum or a Dedekind cut): that is, some operation that guarantees the existence of (or creates, if you like) an object corresponding to a sequence as a whole.

For example, to get a continuum of numbers we assert the existence of a number corresponding to the supremum of every bounded sequence of rationals. To get a collection of ℵ₁ ordinals we assert the existence of an ordinal for the order type of every way of ordering the natural numbers.

[gareth-rees.livejournal.com]

every way of ordering the natural numbers

Of well-ordering the natural numbers is of course what I meant.

(This stuff is tricky to get right. I don't think I've managed one completely correct comment on this topic so far...)

[cartesiandaemon.livejournal.com]

I think that what you're trying to do is impossible:

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 :)

[gareth-rees.livejournal.com]

But am I wrong that, e.g., "put a counter into play for each (possibly infinite) subset of the integers" will be uncountable?

No, you're right.

[alextfish.livejournal.com]

Could you or someone explain to me why ordinals rather than cardinals are what you need? I'm prepared to believe that we don't get to apply 2^(aleph-0) = aleph-1 for some reason, but I don't see why. It seems to me that putting multiple tokens into play isn't something where the order matters (note that Doubling Season's effect doesn't use the stack, it just modifies the rules of the game).

And just for laughs, for a different line of enquiry, you may like to contemplate the possibilities of using Look At Me, I'm R&D to replace 1 with 0, in combination with any card that says "For each 1 ". Such a trick was used in Three Card Blind round 73 to gain a truly infinite amount of life with Renounce, which says "Sacrifice any number of permanents. You gain 2 life for each one sacrificed this way."

[liv]

Edit: This was by me, CartesianDaemon, I forgot to log out.

I'm not sure I grok it all well enough to explain, but I ought to be able to.

2^w and 2^aleph-null are obviously shortcuts. The standard definition for what we'd like from uncountable infinity is not to have a bijection to a aleph-null, since that's how we assigned, eg. assigning blockers. A set of (2 counters 4 counters, 8 counters....) obviously can be counted, ie. have a bijection to 1,2,3... And so isn't what we want, ie. is aleph-0 not aleph-1 or aleph-c. In cardinal arithmetic, 2^x is shorthand for a process which DOES make a "bigger" set, roughly picking one element of two x times, like when defining binary expansions of numbers between 0 and 1.

In ordinal arithmetic, exponentiation represents something different.

A comparison would be addition. Suppose I have infinity counters in play, and add one more. Do I have infinity or infinity plus one? You could argue either way makes sense, but I chose to argue that I have the same number, if I can use them to exactly block an infinite number of creatures. Which is considering the sizes of the sets, ie. cardinal arithmetic, ie. aleph-0 plus 1 is aleph-0 but ω plus 1 is a different number, ω+1.

I agree doubling season is an odd case, because it sort of all happens at once. But I think by any natural interpretation, eg. call the original coutner 1, the counters produced by the first ds 1,2, the counters produced by the next 1,2,3,4... then you'll end up with counters 1,2,3,4... or similar, which after all are plainly countable.

Conversely, if you had an ability that said "put an elf into play for each subset of creatures in play", that would be uncountable, because there's a standard proof that the set of subsets doesn't have a bijection to the original set.

Sorry that wandered a bit, but I hope some of it may be helpful. I may try to evolve (or link to) a more complete justification when I come back.