(First off, let me say right off the bat that I know that canon says some of the planes are infinite. In this case, I'm not talking about what canon says; I acknowledge the fact in the printed material, but this thread is about if the positives of that canon outweigh the negatives. I guess it's kind of like a minor-level PRP for an aspect of the multiverse as a whole or something.)

Talk in the new elemental plane thread got me thinking; is there really a huge reason to make the planes infinite? I'm a mathematician-in-training, and I've got a lot of experience working with infinities. And there are all sorts of unintended side effects as a result of that, especially among the Inner Planes. It seems that the consequences of having something be infinite are often brushed over, or even simply not something the average person would be aware of. On the Outer Planes this stuff seems like less of an issue to me since stuff works so strangely there anyway, so I'll focus mostly on the Inner here. (And if anyone wants to know the specific justification for any of my points, I'll go into it, but I'm trying to avoid actual math unless it becomes necessary.)

For one, it was brought up that Mineral, if it was mostly made up of gems and jewels, would have a huge destabilizing effect on the interplanar economy. People brought up ways to resolve that, including the worthwhile stuff being far less common and the populace of Mineral being so fervant about mining that it didn't happen. But Mineral is infinite. If there's any nonzero percentage of jewels and gems, even something like 1 * 10^-googolplex%, then there's an infinite amount of them. Technically it's even possible there's an infinite amount if there's 0%, though that's harder to illustrate - it's part of the weirdness of infinity. And it's not just mineral. If there's a natural process by which oil could arise in Ooze like some people suggested (an awesome idea, by the way), then there's an infinite amount of oil there, no matter how rare that process is.

That's bad enough, but there's other issues too. First, let's assume that stuff isn't clustered like it seems to be in the material. I'll get to why I'm assuming something demonstrably false by canon after my descriptions, but first I want to go over the consequences of that.

There's issues with distance too. If you take two areas on a given inner plane, the probability of them not being an infinite amount of distance apart is...well, it's hard to define probability of a single event over an infinite space. But it would be literally equally likely to be any real number value apart, as meaningless as that is. Not just the somewhat "local" numbers people are used to, which I'm counting as containing even seemingly tremendous values like the number of particles in the universe, Stokes' number, or other things like that, but literally every real number between 0 and infinity. Even the concept of an average distance is meaningless over an infinite space in a situation like this.

Population too. Are there a finite number of mortal species in the Inner Planes, or no? If there are, then they should be distributed in the same way as the sites I mentioned above, and it should be a literally-0% chance for any two to be near enough to each other so as to have ever met in the history of the multiverse. But if there aren't a finite number of mortal species, where are they getting an infinite amount of resources to survive on in the Inner Planes? Even if they did get an infinite amount of food, that still might not be enough to feed them. Because infinity is very weird. And beyond that, what would the elementals and the mephits (which there should be an infinite number of if the Inner Planes are infinite, I imagine) think of having an infinite number of mortals in their realms?

There's other issues specific to certain planes that I probably haven't even thought of too - one that comes to mind now is that light should either be completely absent or infinitely intense in Air, and it definitely should be infinitely present in Lightning, a la Olbers' paradox - but these are some of the most blatant.

So let's instead go by the canon presentation of having stuff clustered "nearby" one another. Either that means the Inner Planes are packed far more tightly than anyone probably would like them to be, or it means you've got a few million or billion people in a "small" area and an infinite amount of otherwise-empty space. The latter you might be able to pull off, but then the question is "why is it arranged like that". You can put the blame to portal dispersion, but you run into the infinity problem there too. Why are there a finite number of portals? And why aren't they "evenly" distributed throughout the plane, why are they locally clustered? Every time you can push back the explanation a level, all you really do is pull the infinity problem up to another stage.

Now, I'm not going to do this whole spiel without proposing a solution - this was just to explain the problem as I see it with actually using infinity. So how about this?

The infinite planes aren't actually infinite, they're just large 3D surfaces of a set of 4D structures making up the Inner Planes. Or Outer Planes, if you generalized this to that.

For those of you that have read Infinite Staircase, you already know what that means. It's like Maelost. If you go far enough in one direction, eventually you wrap back around to where you started. Not just in 2D, but in 3D even; if you went far enough up, you'd get back to where you started too. And travel from one plane to another could be by natives being able to sense 4D pathways from one plane to another and somehow follow them in 3-space. The planes would then be finite in size, but unbounded. And if you make it large enough - say, Ringworld size, something in the tens-of-thousands-of-planetary-surfaces level - you still get the truly boundless feel to it as well, which is the really important part I think. There are other ways you could do it too - personally, I've always liked thinking of the Hinterlands being a result of the Outlands being a hyperbolic plane, a mathematical construct with an infinite amount of space in a finite circle because the space gets more crunched together the closer you get to the edge of the circle, and maybe the concept of the Hinterlands could be expanded to all the planes the same way.

But anyway, what do the rest of you think? Are there good reasons I'm not seeing beyond "it's canon" to leave the planes as truly infinite, and should these issues just be left ignored? Are there resolutions I'm not seeing to the problems here? I'd appreciate any comments or criticism to this.

Yes the planes have to be infinite.

They aren't supposed to make sense in any realistic fashion. And in my opinion not comprehending that the planes have to physically make sense, is one of the biggest things I hate about some of the 4e design changes made to the planes.