Uncountable infinities in Magic
Jul. 22nd, 2008 03:31 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
jackUpdate, someone else raised a very similar question on a message board. Indeed, using much the same technique I suggested, and I had found that link once before, but forgotten about it.
Page 1
Page 2
On page 1, someone asks a rules question that's a particularly apposite example of uncountable rules:
* The defender has generated an infinite number of small blocking creatures
* (Using a combo which involves mana burning down to a negative infinite amount of life!)
* The attacker has a spell which will win the game if any creature is unblocked
* The attacker uses two Nacatl War Pride, which when it attacks copies itself for each defending creature
* And turns both Nacatl War Pride into creatures that also are doubling season ("whenever a counter is put into play, instead put twice that many into play", although I think only the first one is relevant). Thus the second one puts an infinite number of creatures into play, to which an infinite number of doubling effects apply
* And asks "Will there be any unblocked attacking creatures?"
It's a particularly good example, because the cardinality is exactly relevant: the defender is exactly trying to make a bijection between blocking creatures and attacking creatures, and the attacker wants to know if there will always be an excess attacking creature.
On the second page someone proposes an explicit bijection (or rather, absence of a bijection).
I think this is functionally equivalent to my example.
Page 1
Page 2
On page 1, someone asks a rules question that's a particularly apposite example of uncountable rules:
* The defender has generated an infinite number of small blocking creatures
* (Using a combo which involves mana burning down to a negative infinite amount of life!)
* The attacker has a spell which will win the game if any creature is unblocked
* The attacker uses two Nacatl War Pride, which when it attacks copies itself for each defending creature
* And turns both Nacatl War Pride into creatures that also are doubling season ("whenever a counter is put into play, instead put twice that many into play", although I think only the first one is relevant). Thus the second one puts an infinite number of creatures into play, to which an infinite number of doubling effects apply
* And asks "Will there be any unblocked attacking creatures?"
It's a particularly good example, because the cardinality is exactly relevant: the defender is exactly trying to make a bijection between blocking creatures and attacking creatures, and the attacker wants to know if there will always be an excess attacking creature.
On the second page someone proposes an explicit bijection (or rather, absence of a bijection).
I think this is functionally equivalent to my example.
no subject
Date: 2008-07-22 04:43 pm (UTC)(The defender's choice of partial function may be subject to some kind of restriction: perhaps only recursive functions are allowed? Or maybe since we're allowing infinite numbers of steps, a recursively enumerable -- but not recursive -- function would be acceptable?)
no subject
Date: 2008-07-22 05:52 pm (UTC)That makes sense. Cool.
The defender's choice of partial function may be subject to some kind of restriction:
Now I'm not sure. I'm not sure how soon this would come up if people started trying to construct decks. And I'm not sure if deliberately choosing undecidable or complicated functions is in, or against, the spirit...
My gut reaction is that there ought to be fairly strict restrictions, to approximate what happens in finite magic to some extent. Ideally the blocker would be able to describe how any particular creature blocked or was blocked.
On the other hand, if all the creatures on the same side are the same, and the blocker can demonstrate that he has enough creatures, that feels like it should be sufficient.
Would that ever come up? Can the blocker create an infinite number of creatures, prove they are countable, but not have the production actually produce a bijection, forcing him to fall back on a proof that one exists, but he doesn't have it?
And if I understand rightly, even if the function is primitive recursive, we can still have problems. If blocker #n blockers attacker #n depending on some condition which is checkable for any 'n', but not known -- or even indecidable -- for all n, then we know the answer for any individual blocker, but not what happens in the end. This isn't pretty for playing a game where you know what the result is, but I expect turns out to be unavoidable, but I'm not sure...