“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.
“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.