|
5 years ago ::
Jul 11, 2008 - 12:30PM
#51
|
Date Joined:
Jul 22, 2003
|
I agree with Avedomni that the continuum hypothesis is not required, because I don't care whether it's actually aleph-1, merely that it has a higher order. In that regard, I misspoke when I claimed I meant aleph-1.
I don't know whether raising 2 to the power of beth-0 results in the cardinality of the beth-1, or indeed that it's well-defined at all... but that notation can simply be thought of as an informal shorthand if you prefer.
My actual approach was to construct a bijection between the attacking tokens and the real numbers. Label the doubling season effects D1, D2, etc, in the order that they are applied. Then for each effect's application, when it creates two tokens from one, label those two as D1=0 and D1=1. Then each token has a unique label D1D2D3(...).These labels each map onto a distinct real number between 0 and 1, expressed in binary (0.D1D2D3...), and each real number between 0 and 1 maps onto a label (a token).
Cardinality is defined by bijection relationships, so it seems this would demonstrate the higher cardinality of the attackers.
|
|
|
|
5 years ago ::
Jul 11, 2008 - 12:40PM
#52
|
Date Joined:
Jun 11, 2008
|
|
|
|
|
5 years ago ::
Jul 11, 2008 - 12:46PM
#53
|
Date Joined:
Jul 17, 2001
|
A long time ago, I had a thread about Mox Lotus  in the Unhinged forum. After much debating between very intelligent math majors and the like, it was decided that the concept of infinity simply can't exist within the rules of MTG (no matter what MaRo says...). First of all, there are different types of infinity (which was news to me). Secondly, infinity is a concept, not a number. You can technically only add a number of mana to your mana pool...not a concept. So, it was decided in that thread that the idea of using Mox Lotus in an actual game of Magic, and making rulings around it is a flawed concept from the outset. And that was after much of these same arguments were being presented. I don't think much has changed since then. Casual is Casual. Make up your own rules. Sorry.
|
|
|
|
5 years ago ::
Jul 11, 2008 - 12:47PM
#54
|
Date Joined:
Jun 11, 2008
|
I agree with Avedomni that the continuum hypothesis is not required, because I don't care whether it's actually aleph-1, merely that it has a higher order. In that regard, I misspoke when I claimed I meant aleph-1.
I don't know whether raising 2 to the power of beth-0 results in the cardinality of the beth-1, or indeed that it's well-defined at all... but that notation can simply be thought of as an informal shorthand if you prefer.
My actual approach was to construct a bijection between the attacking tokens and the real numbers. Label the doubling season effects D1, D2, etc, in the order that they are applied. Then for each effect's application, when it creates two tokens from one, label those two as D1=0 and D1=1. Then each token has a unique label D1D2D3(...).These labels each map onto a distinct real number between 0 and 1, expressed in binary (0.D1D2D3...), and each real number between 0 and 1 maps onto a label (a token).
Cardinality is defined by bijection relationships, so it seems this would demonstrate the higher cardinality of the attackers.
While I'm not completely following what's going on here, it seems like it's a more detailed way of saying what I've been trying to say and get across that there will be more attackers than blockers. Is this is case?
|
|
|
|
5 years ago ::
Jul 11, 2008 - 1:19PM
#55
|
- Master of the Quiz (Win '09)
Date Joined:
Apr 23, 2008
|
Not really (well, not in regards to his original post, at least).
In that regard, I misspoke when I claimed I meant aleph-1.
Ah. I see. I was overly pedantic. Please excuse.
While I'm not completely following what's going on here, it seems like it's a more detailed way of saying what I've been trying to say and get across that there will be more attackers than blockers. Is this is case?
sure...?
If you like the advice "Casual is Casual. Make up your own rules." that darkartist gave, then sure.
If both players are casual players, then sure.
If a player can use your explanation and convince his or her opponent, then sure.
If both players take an interest in set theory and have understanding of different transfinite ordinals, then your method of explanation would be insufficient and unsatisfying.
The example NereusRen gave is a good foundation for an argument that has a little more formality.
|
|
|
|
5 years ago ::
Jul 11, 2008 - 1:54PM
#56
|
Date Joined:
Jul 22, 2003
|
While I'm not completely following what's going on here, it seems like it's a more detailed way of saying what I've been trying to say and get across that there will be more attackers than blockers. Is this is case?
That was the goal. If I am right, then there are "more" attackers than blockers, and intuition is similar to your idea of raising something to an infinite power. However, mqj is correct that any explanation that relies on that sort of exponentiation (including my first, where I was trying to avoid getting into too many details about how I reasoned into it) is not meaningful unless you can provide a good definition for that operation. My bijection still isn't a formal proof, but I think it provides enough of an outline that it could be made rigorous and still hold together.
If anyone still thinks that the attackers still have the same cardinality as the blockers, I would challenge them to demonstrate it by finding a bijection between them. A bijection means that they can give me a formula through which, for each attacker, I can determine which creature is assigned to block it.
Since some people seem to be interested in this topic of infinities, I'll give an example of what a bijection would look like:
Say I am attacking with one Nacatl War-Pride and you have infinite blockers. Now I have "one more" attacker than you have blockers. Let's label all my copies as 1, 2, 3, etc., and label your blockers the same way. You can prove that you can block all my attackers by using the following blocking arrangement:
- Blocker 1 blocks the original War-Pride
- Blocker X (for X>1) blocks War-Pride copy X-1.
Once you say that, then there is no attacking creature I can point to that is unblocked. Every attacker is blocked, and you can even say which creature is doing the blocking. This demonstrates that adding 1 to "infinity" doesn't make it any bigger. This method of mapping the two groups to each other 1-to-1 to show an exact match is called a "bijection." For each element of each group, there is a unique one in the other group that you can tie it to. Let's take the case where I have two War-Prides, and construct another bijection. This one will show that "two times infinity" isn't bigger than infinity either. Call the War-Prides A and B, and the copies A1, B1, A2, B2, etc., with the blockers represented by numbers 1, 2, 3, etc., and use the following blocking assignment:
- A blocked by 1
- B blocked by 2
- AX blocked by 2X+1
- BX blocked by 2X+2
You can carry this out a few steps to see that I have not assigned the same blocker to any two attackers, and that all attackers have a blocker assigned to them. When two "different" infinities, like infinity and 2*infinity, have this sort of bijection arrangement between them, they are actually the "same" infinity. In other words, they have the same "cardinality." I hope someone found this simplified explanation interesting  . And thanks again to mqj for your particularly informative replies.
|
|
|
|
5 years ago ::
Jul 12, 2008 - 7:42AM
#57
|
Date Joined:
Jul 16, 2007
|
I just remembered something that could be pretty important to this discussion: If you can generate beth-1 permanents, then why isn't beth-1 itself the amount of mana Mox Lotus gives you? The common assumption seems to be that Mox Lotus generates beth-0 mana, but if we're throwing around multiple kinds of infinities, who are we to say which of those infinities Mox Lotus 's infinite mana symbol actually refers to? Meanwhile...
As you say, the situation's behavior cannot be defined. More specifically, the object to which damage would be dealt is undefined. Of course the rules have nothing to govern this because it cannot come up outside of un-land. If we follow the spirit of the other rules for handling undefined situations, I think it is reasonable to say that the damage is simply not dealt, since there is no way to determine what to deal it to. It's unintuitive in some ways, but consider it an extension of this rule:
"406.6. If a mana ability would produce one or more mana of an undefined type, it produces no mana instead."
Whoa... that's a pretty awesome way to deal with it, actually. I propose: "If a potential action would inevitably be replaced a sequence of replacement effects in a never-ending loop before any actual action could take place, then the original action is impossible." (An exception tailor-made for infinite Doubling Season  combos would probably take anywhere from a paragraph to a whole page to describe.)
While we're on the topic, though, how about this: I use this combo to put an infinite number of Look at Me, I'm R&D  s in play simultaneously, and I specify that one of them cancels out my existing Look at Me, I'm R&D , an infinite number of them just replace 1 with 2, 2 with 1, 3 with 4, 4 with 3, and so forth, and that if the twin prime conjecture is false, another (finite) number of them replace 0 with 1, 1 with 2, 2 with 3, and so on, until they replace p-1 with p, where p is the greatest prime number such that p+2 is prime. (I also make sure their timestamps are in the appropriate order.) This choice of mine is complete and well-defined regardless of whether or not the twin prime conjecture holds. If it does, then the Look at Me, I'm R&D s effectively do nothing, but if it doesn't, then quite a lot of numbers are p now. Do you just set the game aside until somebody determines whether the twin prime conjecture holds?
You would have to assume a couple of things: Continuum Hypothesis and exponentiating aleph-0 with an integer has meaning and would yield aleph-1. Also note that there exists sets with cardinalities larger than the cardinality of the set of real numbers. Those sets are also uncountable.
Well, it seems like you're eventually going to need to accept or reject the continuum hypothesis anyway, as long as you can generate beth-1 permanents. Say my opponent controls beth-0 Crimson Kobolds  and beth-1 Crookshank Kobolds  . I have beth-1 Kobolds of Kher Keep  , but then I sacrifice some number of them to a Renounce  . Can I sacrifice just enough that if I attack with all of my remaining Kobolds of Kher Keep , my opponent can't block them all with just the Crimson Kobolds , but that if my opponent attacks me with all of the Crookshank Kobolds , I can't block them all with the Kobolds of Kher Keep ?
There you go. The continuum hypothesis, in Magic-ese. :P
|
|
|
|
5 years ago ::
Jul 12, 2008 - 8:38AM
#58
|
Date Joined:
May 15, 2001
|
Mox Lotus really needs to be errata'ed to "Add any amount of colorless mana to your mana pool." The concept of infinity is fairly easy. Actually understanding the math behind it, is for math majors and no one else. Asside from using infinity for certain combos (and even then only using it as a limit), there's not much room in the game for it to function correctly. Or, if we're being very rules lawyer-y, we can simply claim "infinity" is undefined and you'd get 0 mana from Mox Lotus. Not in the spirit of the card, but at least it works within the rules. And by the way, once upon a time Magic DID allow for infinity. The rules was that if you were at infinite life, infinite damage would reduce your life total to 0. From that we can assume if you had infinite attacking creatures and I had infinite blocking creatures, I would be able to block them all. This rule was introduced sometime around Ice Age and then they get rid of it around Tempest or 6th Edition. Not sure which.
This is not my sig.
|
|
|
|
5 years ago ::
Jul 12, 2008 - 8:40AM
#59
|
- Celestial Teapots are broken!
Date Joined:
Feb 24, 2007
|
Un- cards don't get errata, and they don't need to work within the rules. It's certainly not the only Un- card that doesn't actually work.
|
|
|
|
5 years ago ::
Jul 12, 2008 - 8:53AM
#60
|
Date Joined:
May 15, 2001
|
The wording I gave is effectivly the same wording on the card, but it allows it to work within the rules. Granted, Un-cards don't always work within the rules set, but very very few have ever caused this kind of confusion.
It'd be used in exactly the same way, but you won't need a math degree to figure out how certain things work. This is a good thing.
This is not my sig.
|
|
|
+ 2(16(∞
