Now, where did the weirdness come from here. Well, to me it seems clear that really it came from the fact that the reals can be built out of a bunch of shifted rational numbers, right?
I think the weirdness comes from trying to assign a real number measure, instead of allowing infinitesimals. I’ve never understood why infinite sets are readily accepted, but infinitesimal/infinite measures are not.
EDIT: To explain my reasoning more, suppose you were Pythagoras and your student came to you and drew a geometric diagram with lengths not in a ratio of whole numbers. You have two options here:
You can declare that not all lengths are commensurate. Not every ratio of lengths results in a number.
You can extend your number system.
Finding the right extension is not an easy problem. Should we extend the numbers to allow square roots (including nesting), but nothing else? This suffices for geometry. But it’s actually more useful to use something like a Cauchy sequence completion: Let any sequence of rational numbers that gets closer and closer together “converge” to a real number. Historically, extending your system of numbers has been what has worked.
When we come across an “immeasurable” set, this to me feels like the same kind of problem. Perhaps we don’t yet have a general consensus on what the “right” extension is to infinitesimals/infinities. However, there clearly are some sets with infinitesimal measure, like the set you constructed. We should figure out a way to give that set infinitesimal measure, not just call it immeasurable.
If you’re talking about surreals or hyperreals, the issue is basically that there’s not one canonical model of infinitesimals, you can create them in many different ways. I’ll hopefully end up writing more about the surreals and hyperreals at some point, but they don’t solve as many issues as you’d hope unfortunately, and actually introduce some other problems.
As a motivating idea here, note that you need the Boolean Prime Ideal Theorem (which requires a weak form of choice to prove) to show that the hyperreals even exist in the first place, if you’re starting from the natural numbers as “mathematically/ontologically basic.” (maybe there’s another way to define them but none immediately come to mind, there is another way to define the surreals, but there are other issues there).
I think the weirdness comes from trying to assign a real number measure, instead of allowing infinitesimals. I’ve never understood why infinite sets are readily accepted, but infinitesimal/infinite measures are not.
EDIT: To explain my reasoning more, suppose you were Pythagoras and your student came to you and drew a geometric diagram with lengths not in a ratio of whole numbers. You have two options here:
You can declare that not all lengths are commensurate. Not every ratio of lengths results in a number.
You can extend your number system.
Finding the right extension is not an easy problem. Should we extend the numbers to allow square roots (including nesting), but nothing else? This suffices for geometry. But it’s actually more useful to use something like a Cauchy sequence completion: Let any sequence of rational numbers that gets closer and closer together “converge” to a real number. Historically, extending your system of numbers has been what has worked.
When we come across an “immeasurable” set, this to me feels like the same kind of problem. Perhaps we don’t yet have a general consensus on what the “right” extension is to infinitesimals/infinities. However, there clearly are some sets with infinitesimal measure, like the set you constructed. We should figure out a way to give that set infinitesimal measure, not just call it immeasurable.
If you’re talking about surreals or hyperreals, the issue is basically that there’s not one canonical model of infinitesimals, you can create them in many different ways. I’ll hopefully end up writing more about the surreals and hyperreals at some point, but they don’t solve as many issues as you’d hope unfortunately, and actually introduce some other problems.
As a motivating idea here, note that you need the Boolean Prime Ideal Theorem (which requires a weak form of choice to prove) to show that the hyperreals even exist in the first place, if you’re starting from the natural numbers as “mathematically/ontologically basic.” (maybe there’s another way to define them but none immediately come to mind, there is another way to define the surreals, but there are other issues there).