View Full Version : Schrodinger's D&D (AKA the metaphysics of roleplaying games)

Ralcos

2015-11-23, 07:36 AM

And now for something completely different...

When I woke up this morning, I never thought I would be trying to justify theoretical quantum physics as part of role-playing games. :smalltongue:

I was thinking of Plane Shift (and similar spells) being used for travelling between alternate universes, and (to make an incredibly long story short) realized that this opens up so many new campaign and gameplay ideas, not just bad Dr. Who fanfiction.

Alternate self as a Cohort (with the Leadership feat)

Cybernetics and/or modern weapons in high fantasy

A collective of alternate universe BBEGs working together as part of the plot

and so many more possibilities...

... Just thought I'd blow your mind with 3 AM ideas and contemplations.

So, what do you guys think?

And just so you know, within my little theorum here, I say that everything is canon and nothing is canon at the same time.

Comet

2015-11-23, 08:32 AM

The module Death Frost Doom (2014 edition) has one room with a bunch of glass balls tucked away in a cupboard or under the bed. These glass balls are actually windows into alternate realities, zoomed in on this same dungeon as it exists in that reality. The players can use a specialised spell to make contact with that alternate reality and speak to the adventurers that are trying to conquer the very same dungeon they are working on at that moment.

The players in our campaign were playing nobodies in a 17th century kind of world with limited magic and even less ability in combat. They contacted one of these alternate realities and found out that, in that world, the dungeon was being raided by a group of warblades, batman wizards, bear totem barbarians and other silly 3.5 superheroes with powerful magics and enchanted gear from head to toe.

Watching that group of what appeared to be gods on earth get annihilated by the dungeon made the players a bit leery about continuing their own adventure.

Beleriphon

2015-11-23, 08:48 AM

<snip> realized that this opens up so many new campaign and gameplay ideas, not just bad Dr. Who fanfiction.

Clearly on the Dr Who note you need a villain that is using time travel to retrieve multiple copies of himself.

Um, it's been done.

Sorry, no blowing of minds here.

Quertus

2015-11-23, 02:24 PM

I was thinking of Plane Shift (and similar spells) being used for travelling between alternate universes, and (to make an incredibly long story short) realized that this opens up so many new campaign and gameplay ideas, not just bad Dr. Who fanfiction.

Alternate self as a Cohort (with the Leadership feat)

Cybernetics and/or modern weapons in high fantasy

A collective of alternate universe BBEGs working together as part of the plot

and so many more possibilities...

... Just thought I'd blow your mind with 3 AM ideas and contemplations.

I'd say that this - the ability to explore alternate realities - is one of the top ten (probably top 3) kinds of game I'd currently like to play. Right up there with "zombie apocalypse" (as something other than "squishy human") and "anything that lets me play some of my old favorite characters".

I've personally done/seen some of what you listed...

Laser guns in high fantasy, when a DM said, "core only", since laser guns are in D&D core

Cybernetics in high fantasy, via D&D grafts (OK, not my character, but a party member)

Self as cohort to alternate self (with leadership feat) - note the slight difference from what you said :smallwink:

A reality-traveling homebrew... where, among other characters, I got to play myself. It was nice to be able to metagame completely in character :smallbiggrin: Less fun was being squishy and hiding (a lot) in a game of "superheroes" :smalleek:

But I'm not sure if Plane Shift, RAW, would be sufficient (although I'd like to think it could work). In... 2e, IIRC... there were specific items that could take the character to alternate game settings, which implied that normal spells were insufficient to do so. One might suspect that, given that there are an infinite number of potential alternate realities, it should be impossible to craft tuning forks for them all without it being more-or-less guaranteed that every tuning fork ever crafted actually takes you to an alternate reality. Does this explode your head?

ExLibrisMortis

2015-11-23, 03:46 PM

There are several infinite planes in the 3.5 cosmology, including the Material Plane. Any infinite plane has an infinite amount of stuff*, including - possibly - an infinite number of adventurers, one of whom might be indistinguishable from your future self. However, there might be more possible adventurers than beings in the Material Plane (despite both being infinite), which would most likely rule out finding a future self. If not, even greater teleport plus a good divination spell will find your double/future/past self.

*This is not actually stated explicitly, but it's implied that you don't cross into 'perfect vacuum from here until infinity, bring food' after a certain amount of space travel.

Kane0

2015-11-23, 04:07 PM

Reminds me of the current BBEG we're chasing down in our game.

He supposedly had this huge army of undead being made and hidden somewhere until he was ready to make his move, but no matter where we looked neither us nor the best casters we had contact with could find this hiding spot.

Turns out it wasn't a where, it was a when. The BBEG wasnt hiding his forces in a place, he was hiding them in a time. Only a 2-3 second distortion, far enough to be considered in the future and thus not present yet close enough that a well place time stop/gate/wish spell could access them if need be.

Clever girl.

Mr.Moron

2015-11-23, 09:11 PM

There are several infinite planes in the 3.5 cosmology, including the Material Plane. Any infinite plane has an infinite amount of stuff*, including - possibly - an infinite number of adventurers, one of whom might be indistinguishable from your future self. However, there might be more possible adventurers than beings in the Material Plane (despite both being infinite), which would most likely rule out finding a future self. If not, even greater teleport plus a good divination spell will find your double/future/past self.

*This is not actually stated explicitly, but it's implied that you don't cross into 'perfect vacuum from here until infinity, bring food' after a certain amount of space travel.

Assume there are an infinite number of infinitely large planes. For any 2x2x2 meter cube of space on any of those planes there is an astronomically high but finite number of ways to arrange the matter in that cube. For any possible arrangement of matter in that cube the chance of that arrangement being "You" as in exactly you down to the subatomic level, is small but by evidence of your existence non-zero. This means that in any given in space in any given plane "You" might be there, and since there are an infinite amount planes and spaces there are infinite number of you.

A properly worded wish spell at the very least, should be able to bring you to one of them. Though has it's own multi-verse collapsing implications. So the fact there is a material plane at all probably implies that the planes and number of planes aren't meaningfully infinite.

Quertus

2015-11-24, 12:21 AM

Assume there are an infinite number of infinitely large planes. For any 2x2x2 meter cube of space on any of those planes there is an astronomically high but finite number of ways to arrange the matter in that cube. For any possible arrangement of matter in that cube the chance of that arrangement being "You" as in exactly you down to the subatomic level, is small but by evidence of your existence non-zero. This means that in any given in space in any given plane "You" might be there, and since there are an infinite amount planes and spaces there are infinite number of you.

A properly worded wish spell at the very least, should be able to bring you to one of them. Though has it's own multi-verse collapsing implications. So the fact there is a material plane at all probably implies that the planes and number of planes aren't meaningfully infinite.

It took me a second to follow your logic. Once I did... OK, head might be ready to explode.

So... assuming an infinite (or infinite x infinite, or infinite ^ 4) amount of space... and a random distribution of matter throughout that space... there would be somewhere between an infinite and an infinite ^ 4 copies of me.

If I cast wish to summon me, then it follows that an infinite number of me simultaneously cast wish to summon me.

However since the me of a second from now and the me of now and the me of a second ago all have slightly different configurations of matter, and since all of these would exist an infinite to infinite ^ 4 times in this universe, it follows that the infinite me's casting wish to summon the infinite me's need not summon the exact same infinite me's that were casting the wish in the first place.

But, even if they didn't summon the infinite number of me's of now, but instead the infinite number of me's of t plus or minus x, that would still summon approximately every me of that time period, which, if their timelines were in any way related, would be very confusing.

... and this is why I try to run the character who was summoned by himself about as often as the character who summoned himself. And why I don't like having a character summon himself repeatedly... unless he is able to exist in multiple places at the same time.

Assume there are an infinite number of infinitely large planes. For any 2x2x2 meter cube of space on any of those planes there is an astronomically high but finite number of ways to arrange the matter in that cube. For any possible arrangement of matter in that cube the chance of that arrangement being "You" as in exactly you down to the subatomic level, is small but by evidence of your existence non-zero. This means that in any given in space in any given plane "You" might be there, and since there are an infinite amount planes and spaces there are infinite number of you.

Without knowing the laws that determine how matter is arranged, the assumption of infinite copies is not justified. It's possible that the multiverse only permits each unique configuration of matter to be used once. If that's the case, then since the number of ways that matter can be arranged is, as you pointed out, finite, then the vast majority of the planes are just empty.

It's also possible that some combinations can occur multiple times, while others can't. So even if the planes are all filled with something, that doesn't necessarily mean there is more than one of you.

ExLibrisMortis

2015-11-24, 10:10 AM

Assume there are an infinite number of infinitely large planes. For any 2x2x2 meter cube of space on any of those planes there is an astronomically high but finite number of ways to arrange the matter in that cube. For any possible arrangement of matter in that cube the chance of that arrangement being "You" as in exactly you down to the subatomic level, is small but by evidence of your existence non-zero. This means that in any given in space in any given plane "You" might be there, and since there are an infinite amount planes and spaces there are infinite number of you.

A properly worded wish spell at the very least, should be able to bring you to one of them. Though has it's own multi-verse collapsing implications. So the fact there is a material plane at all probably implies that the planes and number of planes aren't meaningfully infinite.

You are right when you say that an infinite plane contains an infinite number of copies of every finite combination of matter. You are, however, mistakenly assuming that D&D runs on physical laws that allow only a finite number of ways to arrange matter within a given space. There is no such thing as 'exactly you down to the subatomic level' in D&D, firstly because there are no rules on quantum physics in infinite spaces, and secondly because of magic. In addition, there is no reason to assume adventurers are limited to a certain area, 8 m3 cubes or otherwise - they can be arbitrarily large using cloning powers, for example.

Given a certain ruleset, the cardinality of the set of possible adventurers A might be greater than the cardinality of the set of existing adventurers B, both still being infinite. For instance, a (nonexistant) hivemind composed of all c ∈ C, c ∉ A, C ⊆ B might be considered an adventurer in its own right, giving you A = P(B), giving you |A| > |B|, QED.

Of course, if you're saying that all these composite hiveminds do exist, then I'm flat out of options, but that also means there are so many hiveminds all over the place, it's probably not recognizable or playable as D&D anymore.

Mr.Moron

2015-11-24, 12:02 PM

You are right when you say that an infinite plane contains an infinite number of copies of every finite combination of matter. You are, however, mistakenly assuming that D&D runs on physical laws that allow only a finite number of ways to arrange matter within a given space. There is no such thing as 'exactly you down to the subatomic level' in D&D, firstly because there are no rules on quantum physics in infinite spaces, and secondly because of magic. In addition, there is no reason to assume adventurers are limited to a certain area, 8 m3 cubes or otherwise - they can be arbitrarily large using cloning powers, for example.

Given a certain ruleset, the cardinality of the set of possible adventurers A might be greater than the cardinality of the set of existing adventurers B, both still being infinite. For instance, a (nonexistant) hivemind composed of all c ∈ C, c ∉ A, C ⊆ B might be considered an adventurer in its own right, giving you A = P(B), giving you |A| > |B|, QED.

Of course, if you're saying that all these composite hiveminds do exist, then I'm flat out of options, but that also means there are so many hiveminds all over the place, it's probably not recognizable or playable as D&D anymore.

D&D Doesn't give us rules for a lot of things. That's why wherever the rules don't define a change from ordinary physics and ways of doing things or where the state of the game world doesn't contradict them the only sensible thing to do is assume they're the same as ours. Human beings and other things from the prime material plane, are made of real-worldy matter with atoms.

"Because Magic" isn't a sensible answer unless we're looking to account for something we can't account for without it. Otherwise whenever we go into these "What if.." scenarios the only answer is to throw you arms up in the air and go "Because magic, we don't know. Could be literally anything because any assumptions might be different because magic."

Assuming that matter and it's possible arrangements on the prime material plane work like ours do is a fair one to make for approaching the issue. Others are probably legitimate as well, assuming you can define them consistently enough to actually work with.

Secondly I'm talking about a specific adventurer. The "You" in this case can be considered to be:

Bob

Bob is an 18th-level human wizard from the prime material plane. Bob is 5'6" weighs 170lbs and comfortably fits inside a 2x2x2 cube.

Bob can issue a wish saying

"Teleport me next to another Bob, where "another bob" is defined as another collection of normal matter identical to me. " -or some roughly equivalent statement sufficiently worded to avoid confusion with whatever arbitrary "GM" process is running the wish (assumed to be non-hostile).

EDIT: Even if we say that the number of matter arrangements is infinite, this isn't quite a strong enough statement to say that there aren't multiple identical bobs. We have to be able to say that the chance of the "Bob" arrangement repeating is either zero or follows some kind of arbitrary "Bob Count" limit.

Because in infinite space (even with infinite combinations), the bob combination will still repeat infinitely if the chance of the bob arrangement being in any space other than our reference bob exists at all. We can get more than 1 bob, but less than infinity bobs if the bob arrangement can repeat but there is some rule in the universe saying no more X bobs. However that's getting rather hand-wavey at that point.

ExLibrisMortis

2015-11-24, 05:45 PM

D&D Doesn't give us rules for a lot of things. That's why wherever the rules don't define a change from ordinary physics and ways of doing things or where the state of the game world doesn't contradict them the only sensible thing to do is assume they're the same as ours. Human beings and other things from the prime material plane, are made of real-worldy matter with atoms.

"Because Magic" isn't a sensible answer unless we're looking to account for something we can't account for without it. Otherwise whenever we go into these "What if.." scenarios the only answer is to throw you arms up in the air and go "Because magic, we don't know. Could be literally anything because any assumptions might be different because magic."

Assuming that matter and it's possible arrangements on the prime material plane work like ours do is a fair one to make for approaching the issue. Others are probably legitimate as well, assuming you can define them consistently enough to actually work with.

Secondly I'm talking about a specific adventurer. The "You" in this case can be considered to be:

Bob

Bob is an 18th-level human wizard from the prime material plane. Bob is 5'6" weighs 170lbs and comfortably fits inside a 2x2x2 cube.

Bob can issue a wish saying

"Teleport me next to another Bob, where "another bob" is defined as another collection of normal matter identical to me. " -or some roughly equivalent statement sufficiently worded to avoid confusion with whatever arbitrary "GM" process is running the wish (assumed to be non-hostile).

EDIT: Even if we say that the number of matter arrangements is infinite, this isn't quite a strong enough statement to say that there aren't multiple identical bobs. We have to be able to say that the chance of the "Bob" arrangement repeating is either zero or follows some kind of arbitrary "Bob Count" limit.

Because in infinite space (even with infinite combinations), the bob combination will still repeat infinitely if the chance of the bob arrangement being in any space other than our reference bob exists at all. We can get more than 1 bob, but less than infinity bobs if the bob arrangement can repeat but there is some rule in the universe saying no more X bobs. However that's getting rather hand-wavey at that point.

1) That's right, but since we are talking about infinite planes here, we have no real-world laws to fall back on.

2) It's the only answer we have. D&D magic does not hold with energy conservation, entropy, or blowing up if you stack too much of it in a small space. The existence of this magic neatly invalidates your statement that there is a finite number of possible cubes, because we can - at the very least - always apply another spell to the area.

3) Disagree, because of the above.

4) That's not relevant, really. My statements hold for all adventurers, including Bob (whether he fits in a cube or not).

5 - sidenote) Wishes (in 3.5 anyway) do not require you to word exactly what you want, they just have a verbal spell component :smallwink:.

6) This wish could fail, if the DM rules that there is no such double, based on the tiny chance that such a double exists.

7) Uh, no, that is not how infinities work. I'm saying that there is an infinite space (and infinite adventurers), and a larger infinite set (https://en.wikipedia.org/wiki/Cardinality_of_the_continuum) of possible adventurers. You're right that it is possible to get two Bobs, but the chance is very small, like picking the same real number between 0 and 1 twice in a row. For most adventurers, their copy would not exist (n.b. I'm not a mathematician, though I did some math - I think I'm right, but you never know).

D&D Doesn't give us rules for a lot of things. That's why wherever the rules don't define a change from ordinary physics and ways of doing things or where the state of the game world doesn't contradict them the only sensible thing to do is assume they're the same as ours. Human beings and other things from the prime material plane, are made of real-worldy matter with atoms.

"Because Magic" isn't a sensible answer unless we're looking to account for something we can't account for without it. Otherwise whenever we go into these "What if.." scenarios the only answer is to throw you arms up in the air and go "Because magic, we don't know. Could be literally anything because any assumptions might be different because magic."

Assuming that matter and it's possible arrangements on the prime material plane work like ours do is a fair one to make for approaching the issue. Others are probably legitimate as well, assuming you can define them consistently enough to actually work with.

Secondly I'm talking about a specific adventurer. The "You" in this case can be considered to be:

Bob

Bob is an 18th-level human wizard from the prime material plane. Bob is 5'6" weighs 170lbs and comfortably fits inside a 2x2x2 cube.

Bob can issue a wish saying

"Teleport me next to another Bob, where "another bob" is defined as another collection of normal matter identical to me. " -or some roughly equivalent statement sufficiently worded to avoid confusion with whatever arbitrary "GM" process is running the wish (assumed to be non-hostile).

EDIT: Even if we say that the number of matter arrangements is infinite, this isn't quite a strong enough statement to say that there aren't multiple identical bobs. We have to be able to say that the chance of the "Bob" arrangement repeating is either zero or follows some kind of arbitrary "Bob Count" limit.

Because in infinite space (even with infinite combinations), the bob combination will still repeat infinitely if the chance of the bob arrangement being in any space other than our reference bob exists at all. We can get more than 1 bob, but less than infinity bobs if the bob arrangement can repeat but there is some rule in the universe saying no more X bobs. However that's getting rather hand-wavey at that point.

But if we suppose there is no arbitrary "Bob Count" on the arrangement of matter, what about the non-material aspects of Bob, specifically Bob's soul? Do the Bobs have infinitely many identical souls? If so, wouldn't that get confusing on the Outer Planes? How do Raise Dead or Resurrection spells know which of the infinite identical souls to bring back and reunite with Bob Prime's dead body?

Mr.Moron

2015-11-24, 06:35 PM

But if we suppose there is no arbitrary "Bob Count" on the arrangement of matter, what about the non-material aspects of Bob, specifically Bob's soul? Do the Bobs have infinitely many identical souls? If so, wouldn't that get confusing on the Outer Planes? How do Raise Dead or Resurrection spells know which of the infinite identical souls to bring back and reunite with Bob Prime's dead body?

I don't know and I'm not speculating on that. We have no model for what soul is or does other than it containing some representation of the individuals alignment and personality for upload to the outer planes. Hence why I have bob defining himself strictly in terms of being a collection of matter.

We can't say anything about souls or what they do, unless someone wants to set up some well-defined "Soul Rules" we all agree to follow. I'm not claiming my "Our physics extended out to infinity infinities" approach is the only valid way to approach the problem. Simply that:

A)It is a valid way to approach the problem.

B)It is also the only one we have a consistent, useful model for that we can make meaningful statements about.

If we had different set of rules where we can say "OK Souls do this", "Magic does, this and this and this" fine that's a consistent, useful model we can make meaningful statements about. D&D doesn't really provide any guidelines by default outside the effects narrowly defined by spells, and broad-strokes descriptions of non-material planes.

If somebody wants to put forward a model or models to use other than our universe that's fine and dandy! I'll be happy to work with that. If matter can't be arranged in a finite number of states how does it arrange itself? Are new elements spontaneously generated with unique properties? Is there a uniqueness constraint? Is it strictly probabilistic? Are things composed of something other than atoms in some literal interpretation of ancient alchemy? If so what rules do those structures obey?

Quertus

2015-11-24, 09:43 PM

7) Uh, no, that is not how infinities work. I'm saying that there is an infinite space (and infinite adventurers), and a larger infinite set (https://en.wikipedia.org/wiki/Cardinality_of_the_continuum) of possible adventurers. You're right that it is possible to get two Bobs, but the chance is very small, like picking the same real number between 0 and 1 twice in a row. For most adventurers, their copy would not exist (n.b. I'm not a mathematician, though I did some math - I think I'm right, but you never know).

For the purposes of this experiment, we assume that "most" of the infinite space in the universe has similar conditions; in particular, most of the universe follows the same laws of physics, and is not a black hole.

For a given finite space (2'x2'x2' cube), existing in "normal conditions", there will be between zero and some maximum number of atoms of matter in that space. That number is finite, so any given space has a finite number of atoms.

For any given finite number of atoms, there is a finite number of possible combinations of atoms; ie, 2 atoms could be 2 hydrogen, or 1 hydrogen and 1 helium, or... etc. So there are still a finite number of possible combinations of matter in the space.

(For any given set of atoms, there are a finite number of chemical compounds one can form. This shouldn't matter to the final evaluation.)

Where we get into trouble is where those individual atoms are placed. For each of the three dimensions, there are an infinite number of possible locations for each atom.

So, if, in the above example, we have a subset of space of size infinity ^ 4 (infinite 3-dimensional space, with infinite "planes"). This compared to trying to arrange a given set of atoms, where we have infinity ^ (3 * number of atoms) possible arrangements of atoms.

Sadly, I believe this disproves the infinite Bobs being statistically expected / "guaranteed" in a universe created via random distribution. :smallfrown:

But I'm just going on intuition here. Anyone have a definitive answer?

Raimun

2015-11-25, 01:15 AM

My mind remains unblown.

Plane Shift only lets you travel between Planes. Not through time or dimensions. Teleport only lets you travel between places, within a Plane.

A Plane isn't a different time or an alternate dimension. Even if a Plane is infinite, you wouldn't be able to retrieve your alternate self, because there isn't one to begin with in there. Planes aren't alternate realities like in Quantum Leap, they're just... different from Material Plane. They fill a certain cosmological/mythical need. Some are paradises, some are places of torment, some are on fire, some are nothing but water and some consist of ranch dressing. And just because a D&D-Plane is infinite, doesn't mean it has to contain all the possible possibilities that could exist in the reality. For example, Elemental Plane of Earth is infinite and it's just an infinite amount of earth and rock. If there is anything else beyond that, it's something you brought with you when you Plane Shifted there.

Sure, some obscure third party book probably has dimensional and time travel covered but I'd wager those are best not used, unless you plan to have the campaign to focus on such things.

Satinavian

2015-11-25, 03:26 AM

JoeJ is right.

Just because you have infinite space there is no necessity to have multiples of a certain thing. Because the number of states in a certain part of space (assuming you could actually divide the space into infinite equal parts this way -> assume Euclidian geometry for now) is limited, you wold have multiple copies of something, not neccessarily of everything.

There have to be more assumptions to have multiple copies of everything. Most importantly, that there are no special places in space or time anywhere in the universe. And even then it is fishy because infinity comes without scale and you can have repeating small scale things without repeating larger scale patterns. But in universes you don't have boundaries and everything interacts (in theory) with everything else and thing can be different from each other due to the infinite number of interactions they have with the rest of the universe.

Math with infinities is tricky.

For illustration : Assume a Schwartzschild-metric describes your whole, infinite universe. It's infinite, all right. Still, every volume in space (because time symmetry you can ignore future and past copies) has only a limited number of copies with the same curvature. It's obious, because you have only one point of mass in the whole universe. The universe is still infinite, but there is no reason whatsoever to have any copies of that single point of mass. It's actually even forbidden because it brakes the perfect rotational symmetry of the infinite universe around its very finite number (=1) of centers.

Yes, it's not quantum physics, but in this case it doesn't really make a difference because discussing volumes instead of points and the equation not really caring about what actually is in the center as long as it has the correct symmetries. Even with finite numbers of states instead of continua pretty much everywhere, it is still valid.

It's actually different for parallel universes which are finite each (and have internal quantum physics to get rid of infinite possible states). There you can prove repitition for certain models. And those are, what loosely inspired many of those "parallel universe" stories.

ExLibrisMortis

2015-11-25, 06:45 AM

For the purposes of this experiment, we assume that "most" of the infinite space in the universe has similar conditions; in particular, most of the universe follows the same laws of physics, and is not a black hole.

For a given finite space (2'x2'x2' cube), existing in "normal conditions", there will be between zero and some maximum number of atoms of matter in that space. That number is finite, so any given space has a finite number of atoms.

For any given finite number of atoms, there is a finite number of possible combinations of atoms; ie, 2 atoms could be 2 hydrogen, or 1 hydrogen and 1 helium, or... etc. So there are still a finite number of possible combinations of matter in the space.

(For any given set of atoms, there are a finite number of chemical compounds one can form. This shouldn't matter to the final evaluation.)

Where we get into trouble is where those individual atoms are placed. For each of the three dimensions, there are an infinite number of possible locations for each atom.

So, if, in the above example, we have a subset of space of size infinity ^ 4 (infinite 3-dimensional space, with infinite "planes"). This compared to trying to arrange a given set of atoms, where we have infinity ^ (3 * number of atoms) possible arrangements of atoms.

Sadly, I believe this disproves the infinite Bobs being statistically expected / "guaranteed" in a universe created via random distribution. :smallfrown:

But I'm just going on intuition here. Anyone have a definitive answer?

I think your intuition is correct, but some of your assumptions are less than complete. I just want to add a few things.

As I said before, there is an infinite number of possible cubes in D&D, because magic is not atomic and can be layered ad infinitum, and transmutation spells can do weird things to atoms, too (e.g. what do you do when matter manipulation increases the hardness of adamantine, obdurium or tungsten carbide?). Outside D&D, you could argue that a gravitational singularity (the centre of a black hole) fits in a cube, so that you can always increase the mass of the heaviest singularity to get a new heaviest black hole, leading to an infinite number of possible cubes (most of which are cubes with singularities).

In addition, space may be quantized (as in loop quantum gravity), resulting in a finite number of arrangements of matter in a given cube in Real Lifetm. We don't actually know how that works, so there's no real way to implement that in D&D. For now, we can assume infinitely many arrangements, however, due to the uncertainty principle, we would be unable to measure the difference between certain cubes, at least, barring the use of magic (which is again a wild card that shuts down arguments everywhere :smalltongue:).

We don't have an infinite number of infinite planes, sadly (there are only about thirty, in the standard cosmology, variant cosmologies can change that, but not here). However, that probably doesn't change your argument, as we have the same amount of space available.

Infinity (that is, an infinite cardinal, not the symbol ∞) to the power of some number equals said infinity. Two (or any number >1) to the power of some infinity equals a larger infinity. You are taking the limit at infinity, which does allow you to take x4/x3 + y and call it 0 at infinity for y > 1, but that doesn't work for actual infinities.

The number of possible arrangements of matter in 3d-space is probably something like 2|ℝ|. That's a greater infinity than |ℝ3|. Based on some quick searching, the number of permutations of an infinite set of cardinality K is K!, which equals 2K (roughly speaking, there are more ways to arrange a set of numbers than there are numbers).

Satinavian

2015-11-25, 08:01 AM

In addition, space may be quantized (as in loop quantum gravity), resulting in a finite number of arrangements of matter in a given cube in Real Lifetm. We don't actually know how that works, so there's no real way to implement that in D&D. For now, we can assume infinitely many arrangements, however, due to the uncertainty principle, we would be unable to measure the difference between certain cubes, at least, barring the use of magic (which is again a wild card that shuts down arguments everywhere :smalltongue:).I think, quantized space is assumed. Otherwise the whole argument of a finite number of arrangements in a certain space would have been wrong from the start.

Infinity (that is, an infinite cardinal, not the symbol ∞) to the power of some number equals said infinity. Two (or any number >1) to the power of some infinity equals a larger infinity. You are taking the limit at infinity, which does allow you to take x4/x3 + y and call it 0 at infinity for y > 1, but that doesn't work for actual infinities.

The number of possible arrangements of matter in 3d-space is probably something like 2|ℝ|. That's a greater infinity than |ℝ3|. Based on some quick searching, the number of permutations of an infinite set of cardinality K is K!, which equals 2K (roughly speaking, there are more ways to arrange a set of numbers than there are numbers).If you really really want to those things, you have to distinguish between different kinds of infinity, at least using set theory a la Cantor. Some of your equations are true for one kind of infinity but not for another.

Which brings us back to the quantization of spacetime. Which completely changes the kind of infinities we have to consider.

FlumphPaladin

2015-11-25, 10:51 AM

Now to convince the team to roll up a party consisting of alternate-universe versions of the same person.

ExLibrisMortis

2015-11-25, 10:52 AM

If you really really want to those things, you have to distinguish between different kinds of infinity, at least using set theory a la Cantor. Some of your equations are true for one kind of infinity but not for another.

Err, I am distinguishing between different infinities. My equations are correct. What part do you think is wrong, and why?

Satinavian

2015-11-25, 01:13 PM

Err, I am distinguishing between different infinities. My equations are correct. What part do you think is wrong, and why?

Sorry, actually misread something. You are indead correct with the equations themself.

But i don't think, you have the correct infinities to describe the problem at hand. It never occured to me to assume the number of states in a given volume to be more than aleph-naugh or aleph-one at most if infinite at all. Are you doing classical (= non quantum ) physics here ?

ExLibrisMortis

2015-11-25, 04:51 PM

Sorry, actually misread something. You are indead correct with the equations themself.

But i don't think, you have the correct infinities to describe the problem at hand. It never occured to me to assume the number of states in a given volume to be more than aleph-naugh or aleph-one at most if infinite at all. Are you doing classical (= non quantum ) physics here ?

Well, here's the thing.

In quantized spacetime, I'd agree there is a finite number of arrangements of matter. However, the quantization of space is not 100% confirmed (it's not part of general relativity, as far as I know, which is causing some clashes with quantum mechanics), therefore I wasn't assuming it. However, even if I was, we get to point two.

We are discussing a D&D setting, and in D&D, you are dealing with magic. In that sense, I'm dealing with 'non-quantum' physics here. In D&D, you can layer about as many mage armour spells onto someone as you want. Naturally, you get only one bonus, but each one must be dispelled separately. Clearly, an adventurer - or a cube containing said adventurer - with 1.000.001 mage armours active is different from an otherwise identical adventurer with 1.000.002 mage armours active. That way, I'm proving that there are at least ℕ adventurers (although not ℝ, sadly).

As for the cardinality of the set of possible arrangements of matter being greater than |ℝ|, well, I just googled it, really. The number of pertubations of an infinite set A is 2|A|, apparently. Since the set of points S within a 2x2x2 metre cube has a cardinality ℝ (this is again based on non-quantized space), this means there must be 2|ℝ| pertubations, greater than |ℝ|.

Edit: sorry, that assumes I have ℝ unique points, which isn't neccessarily the case, even if there are ℝ points. However, I came up with an alternate way of doing it: stacking sentient swarms with damage reduction. Swarms can enter eachothers spaces, sentience ensures they all have different personalities (and they're also potential adventurers, for that matter), and the damage reduction makes them immune to the damage that swarms do to creatures in their space.

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