There are measurable sets whose volumes will not be preserved if you try to measure them with surreal numbers. For example, consider [0,∞)⊆R. Say its measure is some infinite surreal number n. The volume-preserving left-shift operation x↦x−1 sends [0,∞) to [−1,∞), which has measure 1+n, since [−1,0) has measure 1. You can do essentially the same thing in higher dimensions, and the shift operation in two dimensions ((x,y)↦(x−1,y)) can be expressed as the composition of two rotations, so rotations can’t be volume-preserving either. And since different rotations will have to fail to preserve volumes in different ways, this will break symmetries of the plane.
I wouldn’t say that volume-preserving transformations fail to preserve volume on non-measurable sets, just that non-measurable sets don’t even have measures that could be preserved or not preserved. Failing to preserve measures of sets that you have assigned measures to is entirely different. Non-measurable sets also don’t arise in mathematical practice; half-spaces do. I’m also skeptical of the existence of non-measurable sets, but the non-existence of non-measurable sets is a far bolder claim than anything else I’ve said here.
I’m happy to bite that bullet and destroy the symmetry. If we pick a random point and line in the universe, are there more unit points to the left or right? Well, that depends on where the point is.
It’s a bad bullet to bite. Its symmetries are essential to what makes Euclidean space interesting.
And here’s another one: are you not bothered by the lack of countable additivity? Suppose you say that the volume of Euclidean space is some surreal number n. Euclidean space is the union of an increasing sequence of balls. The volumes of these balls are all finite, in particular, less than n2, so how can you justify saying that their union has volume greater than n2?
“Its symmetries are essential to what makes Euclidean space interesting”—Isn’t the interesting aspect of Euclidean space its ability to model our world excluding relativity?
Well, I just don’t think it’s that unusual for functions to have properties that break at their limits. Is this any different from 1/x being definable everywhere except 0? Is there anything that makes the change at the limit particularly concerning.
I don’t follow the analogy to 1/x being a partial function that you’re getting at.
Maybe a better way to explain what I’m getting at is that it’s really the same issue that I pointed out for the two-envelopes problem, where you know the amount of money in each envelope is finite, but the uniform distribution up to an infinite surreal would suggest that the probability that the amount of money is finite is infinitesimal. Suppose you say that the size of the ray [0,∞) is an infinite surreal number n. The size of the portion of this ray that is distance at least r from 0 is n−r when r is a positive real, so presumably you would also want this to be so for surreal r. But using, say, r:=√n, every point in [0,∞) is within distance √n of 0, but this rule would say that the measure of the portion of the ray that is farther than √n from 0 is n−√n; that is, almost all of the measure of [0,∞) is concentrated on the empty set.
There are measurable sets whose volumes will not be preserved if you try to measure them with surreal numbers. For example, consider [0,∞)⊆R. Say its measure is some infinite surreal number n. The volume-preserving left-shift operation x↦x−1 sends [0,∞) to [−1,∞), which has measure 1+n, since [−1,0) has measure 1. You can do essentially the same thing in higher dimensions, and the shift operation in two dimensions ((x,y)↦(x−1,y)) can be expressed as the composition of two rotations, so rotations can’t be volume-preserving either. And since different rotations will have to fail to preserve volumes in different ways, this will break symmetries of the plane.
I wouldn’t say that volume-preserving transformations fail to preserve volume on non-measurable sets, just that non-measurable sets don’t even have measures that could be preserved or not preserved. Failing to preserve measures of sets that you have assigned measures to is entirely different. Non-measurable sets also don’t arise in mathematical practice; half-spaces do. I’m also skeptical of the existence of non-measurable sets, but the non-existence of non-measurable sets is a far bolder claim than anything else I’ve said here.
Well shifting left produces a superset of the original, so of course we shouldn’t expect that to preserve measure.
What about rotations, and the fact that we’re talking about destroying a bunch of symmetry of the plane?
I’m happy to bite that bullet and destroy the symmetry. If we pick a random point and line in the universe, are there more unit points to the left or right? Well, that depends on where the point is.
It’s a bad bullet to bite. Its symmetries are essential to what makes Euclidean space interesting.
And here’s another one: are you not bothered by the lack of countable additivity? Suppose you say that the volume of Euclidean space is some surreal number n. Euclidean space is the union of an increasing sequence of balls. The volumes of these balls are all finite, in particular, less than n2, so how can you justify saying that their union has volume greater than n2?
“Its symmetries are essential to what makes Euclidean space interesting”—Isn’t the interesting aspect of Euclidean space its ability to model our world excluding relativity?
Well, I just don’t think it’s that unusual for functions to have properties that break at their limits. Is this any different from 1/x being definable everywhere except 0? Is there anything that makes the change at the limit particularly concerning.
I don’t follow the analogy to 1/x being a partial function that you’re getting at.
Maybe a better way to explain what I’m getting at is that it’s really the same issue that I pointed out for the two-envelopes problem, where you know the amount of money in each envelope is finite, but the uniform distribution up to an infinite surreal would suggest that the probability that the amount of money is finite is infinitesimal. Suppose you say that the size of the ray [0,∞) is an infinite surreal number n. The size of the portion of this ray that is distance at least r from 0 is n−r when r is a positive real, so presumably you would also want this to be so for surreal r. But using, say, r:=√n, every point in [0,∞) is within distance √n of 0, but this rule would say that the measure of the portion of the ray that is farther than √n from 0 is n−√n; that is, almost all of the measure of [0,∞) is concentrated on the empty set.