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Many of the comments here are critical of Cantor or his proofs… it’s reminiscent of the reaction that the poor guy received from his contemporaries, who basical

Many of the comments here are critical of Cantor or his proofs… it’s reminiscent of the reaction that the poor guy received from his contemporaries, who basically shunned him for many years. It’s a fundamental mathematical fact that there does not exist a bijection between the naturals and the reals. This is what provides genuine distinction between the discrete and the continuous, the countable and the uncountable. It certainly is a mind-blowing fact. But many facts blow the human mind, and it isn’t Cantor’s fault that he was brilliant enough to discover one such.

xscott 5y ago
There is however a bijection between the natural/countable numbers and every number that mathematicians, physicists, and any other scientist have every used in all of human history (or will ever use). Start with the computable numbers, and then generalize to anything that has been written down in symbolic text. It's very hard to point to a real number which is not in the set of computable numbers, and for the few examples (Chaitin's constant(s) etc...), there are countably many of them. From this point of view, I'm curious whether the computable numbers are sufficient to do integral and differential calculus etc... (basically all "normal" math that engineers or applied scientists might use) Maybe it requires a different definition of limits, I dunno. What number are we interested in that can't be done with computable numbers? At what point do we need Real numbers and their mind-blowingly weird properties?
- adenadel 5y ago
  I am certainly not an expert on this, but my guess is that there will be be challenges to doing calculus since you won't have the standard foundational properties of the reals (like the least upper bound property). That being said, it looks like mathematicians have investigated this.[0] 0. https://en.wikipedia.org/wiki/Computable_analysis https://en.wikipedia.org/wiki/Computable_analysis
- dooglius 5y ago
  It's not individual numbers that matter, but the properties of sets of them and functions over them. You can define a Reimann integral over computable numbers without changing much, but not the more general Lebesgue integral. This means you won't get e.g. the nice properties of L2 spaces which are used heavily in quantum mechanics.
  - xscott 5y ago
    I'm not sure if you'll see this reply since it's been several days, but I appreciate the answer. I majored in math, but I never stumbled on the Lebesgue integral, and I don't know enough quantum to understand the subtlety there either. I'll have to take your word for it that L2 spaces don't work with just computable numbers, but I can't see why yet. All of that to say, I'm just admitting my ignorance and not arguing at all. Still, I think it's interesting that if Reals are truly needed to model the universe (physics), then there must be some case where computables are insufficient. And if the universe isn't computable, that says something amusing and deep about determinism and whether we can ever succeed in modelling it. :-)
    - dooglius 5y ago
      One of the major pillars of measure theory is countable subadditivity, which serves to allow one to work with countably infinite series in a nice way. This is pretty much unworkable in a countable space like the computables. That isn't to say we really "need" the reals, merely that math hasn't come up with a fully-featured alternative framework that physics can use. I know there are people working on "computable measure theory" in recent years, and there's no reason why the universe should use a mathematical framework that is easy for we humans to construct.
- chobytes 5y ago
  A fun wrinkle is that there are (infinitely many) models of set theory which are pointwise definable. Thus in the right model of zfc, every mathematical object is uniquely addressable by a certain finite formula. (see https://arxiv.org/pdf/1105.4597.pdf https://arxiv.org/pdf/1105.4597.pdf )
ASalazarMX 5y ago
I think it's mostly a distinction between abstract and practical concepts. You might say that the infinite set of real numbers between 0 and 1 is as big as the infinite set of positive integers, but that's just true in an abstract sense. In the practical sense, for example, there are not infinite frequencies between 0 and 1 Hertz because they're subject to physical limits.
- anikan_vader 5y ago
  > You might say that the infinite set of real numbers between 0 and 1 is ad big as [N] No, I would say that it is much bigger, since [0, 1] is uncountable. > there are not infinite frequencies between 0 and 1. Personally I think time is continuous. It sounds like you believe it is discrete. But such a debate is entirely irrelevant to the set theory.
  - ASalazarMX 5y ago
    And that is another example of the abstract vs pragmatic visions: it hasn't been proved that time is (or isn't) quantized, so we're left with out own opinions. Seeing that time needs things happening, and things are made of matter, which is quantized, I'm inclined to think time is quantized too.

xscott 5y ago

There is however a bijection between the natural/countable numbers and every number that mathematicians, physicists, and any other scientist have every used in all of human history (or will ever use). Start with the computable numbers, and then generalize to anything that has been written down in symbolic text. It's very hard to point to a real number which is not in the set of computable numbers, and for the few examples (Chaitin's constant(s) etc...), there are countably many of them. From this point of view, I'm curious whether the computable numbers are sufficient to do integral and differential calculus etc... (basically all "normal" math that engineers or applied scientists might use) Maybe it requires a different definition of limits, I dunno. What number are we interested in that can't be done with computable numbers? At what point do we need Real numbers and their mind-blowingly weird properties?

adenadel 5y ago
I am certainly not an expert on this, but my guess is that there will be be challenges to doing calculus since you won't have the standard foundational properties of the reals (like the least upper bound property). That being said, it looks like mathematicians have investigated this.[0] 0. https://en.wikipedia.org/wiki/Computable_analysis https://en.wikipedia.org/wiki/Computable_analysis
dooglius 5y ago
It's not individual numbers that matter, but the properties of sets of them and functions over them. You can define a Reimann integral over computable numbers without changing much, but not the more general Lebesgue integral. This means you won't get e.g. the nice properties of L2 spaces which are used heavily in quantum mechanics.
- xscott 5y ago
  I'm not sure if you'll see this reply since it's been several days, but I appreciate the answer. I majored in math, but I never stumbled on the Lebesgue integral, and I don't know enough quantum to understand the subtlety there either. I'll have to take your word for it that L2 spaces don't work with just computable numbers, but I can't see why yet. All of that to say, I'm just admitting my ignorance and not arguing at all. Still, I think it's interesting that if Reals are truly needed to model the universe (physics), then there must be some case where computables are insufficient. And if the universe isn't computable, that says something amusing and deep about determinism and whether we can ever succeed in modelling it. :-)
  - dooglius 5y ago
    One of the major pillars of measure theory is countable subadditivity, which serves to allow one to work with countably infinite series in a nice way. This is pretty much unworkable in a countable space like the computables. That isn't to say we really "need" the reals, merely that math hasn't come up with a fully-featured alternative framework that physics can use. I know there are people working on "computable measure theory" in recent years, and there's no reason why the universe should use a mathematical framework that is easy for we humans to construct.
chobytes 5y ago
A fun wrinkle is that there are (infinitely many) models of set theory which are pointwise definable. Thus in the right model of zfc, every mathematical object is uniquely addressable by a certain finite formula. (see https://arxiv.org/pdf/1105.4597.pdf https://arxiv.org/pdf/1105.4597.pdf )

ASalazarMX 5y ago

I think it's mostly a distinction between abstract and practical concepts. You might say that the infinite set of real numbers between 0 and 1 is as big as the infinite set of positive integers, but that's just true in an abstract sense. In the practical sense, for example, there are not infinite frequencies between 0 and 1 Hertz because they're subject to physical limits.

anikan_vader 5y ago
> You might say that the infinite set of real numbers between 0 and 1 is ad big as [N] No, I would say that it is much bigger, since [0, 1] is uncountable. > there are not infinite frequencies between 0 and 1. Personally I think time is continuous. It sounds like you believe it is discrete. But such a debate is entirely irrelevant to the set theory.
- ASalazarMX 5y ago
  And that is another example of the abstract vs pragmatic visions: it hasn't been proved that time is (or isn't) quantized, so we're left with out own opinions. Seeing that time needs things happening, and things are made of matter, which is quantized, I'm inclined to think time is quantized too.