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> the mathematical universe, like our physical one, may be made up mostly of dark matter. “It seems now that most of the universe somehow consists of things tha

> the mathematical universe, like our physical one, may be made up mostly of dark matter. “It seems now that most of the universe somehow consists of things that we can’t see,” Not heaps fond of relating invisible things in the mathematical universe to dark matter! Although maybe both might turn out to be imaginary/purely-abstract? Imaginary things can absolutely influence real things in the universe, it's just that they are not usually external to the thing they are influencing. If I imagine making a cake say, and then I go ahead and make the one I imagined, the 'virtual' cake was already inside me to begin with, and wasn't 'plucked' from a virtual universe of possible cakes somewhere outside my knowledge of cake-making. Something nags at the back of my mind around this about maths though, as if to suggest that as soon as there was one-of-anything that was kinda an 'instantiation' of the most abstract "one" object from the mathematical universe.. (irrespective of what axioms are used as long as they support something like one) But I doubt there's never been exactly-PI-of-anything in the real universe, just a whole bunch of systems that behave as if they know (or are perhaps in the process of computing) a more exact value! (spherical planets, natural sine waves etc!) Very interesting article, I wish my math was stronger! I can just skirt the edges of what they're actually talking about and it's tantalizing! Would love to know more about these new types of cardinal numbers they've developed/discovered.

cyborgx7 1y ago
An interesting thing about the quote you highlighted is that it's already true about the set of real numbers itself. The set of real numbers that can be precisely, individually identified is a countable subset of all real numbers. That means the vast majority of real numbers, an uncountable amount of them, can not be individually defined and thought about.
- danwills 1y ago
  That is very interesting I agree, and certainly any list of descriptions/identifiers must be countable, though I wonder if there's any validity in descriptions that describe things in aggregate? It's certainly a brain-bender that even in the unit interval if we imagine filling in all the the rationals and then adding in the describable-irrationals like PI/4, sqrt(2)/2 and so on.. that this still does not even come close to covering the unit interval - or any interval - of Real numbers! My imagination sees a line with a heck of a lot of dots on it, but still knowing that there clearly still uncountably-more values that are not covered/described! Amazing! The continuum (Real numbers) is such a fascinating concept!
  - nyrikki 1y ago
    Almost all real numbers are normal numbers, which don't even have a finite representation. Sure you can assign them to an arbitrary set, but you don't have access to the value. It is a hay in the haystack problem, where you really only have access to the needles, not the hay.
    - gowld 1y ago
      It's even more extreme than that! Take the (uncountable) set of Real numbers. Remove the normal numbers, which is almost all of them in the sense that the probability that "a uniformly randomly chosen real number is normal (and therefore also undescribable)" is 1. The remaining set of numbers, which has measure 0 in the Real numbers, is still uncountable, meaning that the proability of randomly choosing a describable number in that set is again 0. I'm not sure how deep this chain can go. Google AI says "only 1 steps" but it's not admiting the case described in this comment.
    - LegionMammal978 1y ago
      > Almost all real numbers are normal numbers, which don't even have a finite representation. Plenty of normal numbers have a finite representation from which digits can be efficiently extracted. E.g., Champernowne's constant (in any base) is normal, and you can find its digits with a relatively simple algorithm. All computable reals can similarly have their digits extracted by some algorithm or another, even though it may take a long time. I wouldn't call that "not having access to the value". Of course, uncomputable numbers are a different story, but they have nothing to do with normality in any base. And of course, radix representations are not the only way to evaluate real numbers. E.g., you could represent them with simple continued fractions (which would still allow addition, multiplication, comparison, etc.), and then you could write out any quadratic irrational with a periodic expansion.
      - nyrikki 1y ago
        "almost all" or "almost everywhere" is in italics because I mean it in the measure theory sense. Meaning it holds for all elements of a set except for a subset that has measure zero. Yes some normal numbers are in the constructable reals, but it is a measure zero subset. You are putting your hand in the haystack and only finding needles, finding the hay in the haystack is the problem here.
- dwohnitmok 1y ago
  This is subtle and a simple counting argument (definable means satisfies a finite formula, there are only countably many finite formulas, there are uncountably many reals, therefore there must be undefinable reals) doesn't work, because "definable in ZFC" is not something that is formalizable in ZFC and so the usual set-theoretic counting arguments don't work. So it is in fact possible and consistent with ZFC that all reals are definable. See: https://mathoverflow.net/questions/44102/is-the-analysis-as-taught-in-universities-in-fact-the-analysis-of-definable-numb/44129#44129 https://mathoverflow.net/questions/44102/is-the-analysis-as-...
  - bubblyworld 1y ago
    Thanks, that's a wonderful link and a nice puzzle to think about. The best intuition I have for it is that since the predicate "isDefinableReal(x)" is not itself definable in first-order set theory, there is no way to construct the set of all definable reals in the first place. Thus saying it's countable is basically meaningless - what, exactly, is countable?
    - drdeca 1y ago
      If you use ZFC+Consistent(ZFC) as your meta-theory, and within it consider a model of ZFC, then surely one can consider the set (in the meta theory) of sentences which pick out a unique real number in the model, and then the set of real numbers in the model which are picked out by some sentence? It might not be a set that belongs to the model, but it’s a set in the meta-theory, right? And, I imagine that the set of real numbers of the meta theory could be (in the meta theory) the same set as the set of real numbers in the model?
      - bubblyworld 1y ago
        You can do this, but things get strange in the meta-theory. Some models of ZFC are countable according to the meta-theory! And some of them have models of the reals that are countable according to the meta-theory. There's no contradiction here, because what the meta-theory thinks "countable" means has nothing to do with what the inner model thinks "countable" means. (for an extreme example of this, by the Löwenheim–Skolem theorem there are countable models of ZFC) So you can do what you are suggesting, and you will of course get a countable set of reals (or what are reals according to the inner model), but they might not be countable according to the inner model. They might not even be a set according to the inner model, and there are even inner models that think you've got all of the reals! (see https://mathoverflow.net/questions/351659/set-of-definable-real-numbers https://mathoverflow.net/questions/351659/set-of-definable-r... pretty heavy reading) So the statement "the set of definable reals is countable" is nonsense - you're talking about things that live in different universes of meaning.
- gbacon 1y ago
  My intuition about the question is related: the set of all Turing machines (algorithms) is countable, but the set of all languages (problems to solve) is uncountable. If you take mathematics to be the bigger, uncountable picture, it’s mostly chaos, but if you limit consideration to algorithms, then it’s mostly order.
  - Viliam1234 1y ago
    > the set of all languages (problems to solve) is uncountable The set of all problems that can be described by a finite description is countable. Why would we care about the rest of them?
    - jibal 1y ago
      Because we're theorists.
- seanhunter 1y ago
  That is not true. Algebraic numbers for example are an uncountably infinite subset of the real numbers that can be precisely individually identified.
  - cyborgx7 1y ago
    According to Wikipedia, there are countably many algebraic numbers, and that makes intuitive sense to me as well. Do you have a source that the set of algebraic numbers is uncountable?
    - seanhunter 1y ago
      I am probably wrong

cyborgx7 1y ago

An interesting thing about the quote you highlighted is that it's already true about the set of real numbers itself. The set of real numbers that can be precisely, individually identified is a countable subset of all real numbers. That means the vast majority of real numbers, an uncountable amount of them, can not be individually defined and thought about.

danwills 1y ago
That is very interesting I agree, and certainly any list of descriptions/identifiers must be countable, though I wonder if there's any validity in descriptions that describe things in aggregate? It's certainly a brain-bender that even in the unit interval if we imagine filling in all the the rationals and then adding in the describable-irrationals like PI/4, sqrt(2)/2 and so on.. that this still does not even come close to covering the unit interval - or any interval - of Real numbers! My imagination sees a line with a heck of a lot of dots on it, but still knowing that there clearly still uncountably-more values that are not covered/described! Amazing! The continuum (Real numbers) is such a fascinating concept!
- nyrikki 1y ago
  Almost all real numbers are normal numbers, which don't even have a finite representation. Sure you can assign them to an arbitrary set, but you don't have access to the value. It is a hay in the haystack problem, where you really only have access to the needles, not the hay.
  - gowld 1y ago
    It's even more extreme than that! Take the (uncountable) set of Real numbers. Remove the normal numbers, which is almost all of them in the sense that the probability that "a uniformly randomly chosen real number is normal (and therefore also undescribable)" is 1. The remaining set of numbers, which has measure 0 in the Real numbers, is still uncountable, meaning that the proability of randomly choosing a describable number in that set is again 0. I'm not sure how deep this chain can go. Google AI says "only 1 steps" but it's not admiting the case described in this comment.
  - LegionMammal978 1y ago
    > Almost all real numbers are normal numbers, which don't even have a finite representation. Plenty of normal numbers have a finite representation from which digits can be efficiently extracted. E.g., Champernowne's constant (in any base) is normal, and you can find its digits with a relatively simple algorithm. All computable reals can similarly have their digits extracted by some algorithm or another, even though it may take a long time. I wouldn't call that "not having access to the value". Of course, uncomputable numbers are a different story, but they have nothing to do with normality in any base. And of course, radix representations are not the only way to evaluate real numbers. E.g., you could represent them with simple continued fractions (which would still allow addition, multiplication, comparison, etc.), and then you could write out any quadratic irrational with a periodic expansion.
    - nyrikki 1y ago
      "almost all" or "almost everywhere" is in italics because I mean it in the measure theory sense. Meaning it holds for all elements of a set except for a subset that has measure zero. Yes some normal numbers are in the constructable reals, but it is a measure zero subset. You are putting your hand in the haystack and only finding needles, finding the hay in the haystack is the problem here.
dwohnitmok 1y ago
This is subtle and a simple counting argument (definable means satisfies a finite formula, there are only countably many finite formulas, there are uncountably many reals, therefore there must be undefinable reals) doesn't work, because "definable in ZFC" is not something that is formalizable in ZFC and so the usual set-theoretic counting arguments don't work. So it is in fact possible and consistent with ZFC that all reals are definable. See: https://mathoverflow.net/questions/44102/is-the-analysis-as-taught-in-universities-in-fact-the-analysis-of-definable-numb/44129#44129 https://mathoverflow.net/questions/44102/is-the-analysis-as-...
- bubblyworld 1y ago
  Thanks, that's a wonderful link and a nice puzzle to think about. The best intuition I have for it is that since the predicate "isDefinableReal(x)" is not itself definable in first-order set theory, there is no way to construct the set of all definable reals in the first place. Thus saying it's countable is basically meaningless - what, exactly, is countable?
  - drdeca 1y ago
    If you use ZFC+Consistent(ZFC) as your meta-theory, and within it consider a model of ZFC, then surely one can consider the set (in the meta theory) of sentences which pick out a unique real number in the model, and then the set of real numbers in the model which are picked out by some sentence? It might not be a set that belongs to the model, but it’s a set in the meta-theory, right? And, I imagine that the set of real numbers of the meta theory could be (in the meta theory) the same set as the set of real numbers in the model?
    - bubblyworld 1y ago
      You can do this, but things get strange in the meta-theory. Some models of ZFC are countable according to the meta-theory! And some of them have models of the reals that are countable according to the meta-theory. There's no contradiction here, because what the meta-theory thinks "countable" means has nothing to do with what the inner model thinks "countable" means. (for an extreme example of this, by the Löwenheim–Skolem theorem there are countable models of ZFC) So you can do what you are suggesting, and you will of course get a countable set of reals (or what are reals according to the inner model), but they might not be countable according to the inner model. They might not even be a set according to the inner model, and there are even inner models that think you've got all of the reals! (see https://mathoverflow.net/questions/351659/set-of-definable-real-numbers https://mathoverflow.net/questions/351659/set-of-definable-r... pretty heavy reading) So the statement "the set of definable reals is countable" is nonsense - you're talking about things that live in different universes of meaning.
gbacon 1y ago
My intuition about the question is related: the set of all Turing machines (algorithms) is countable, but the set of all languages (problems to solve) is uncountable. If you take mathematics to be the bigger, uncountable picture, it’s mostly chaos, but if you limit consideration to algorithms, then it’s mostly order.
- Viliam1234 1y ago
  > the set of all languages (problems to solve) is uncountable The set of all problems that can be described by a finite description is countable. Why would we care about the rest of them?
  - jibal 1y ago
    Because we're theorists.
seanhunter 1y ago
That is not true. Algebraic numbers for example are an uncountably infinite subset of the real numbers that can be precisely individually identified.
- cyborgx7 1y ago
  According to Wikipedia, there are countably many algebraic numbers, and that makes intuitive sense to me as well. Do you have a source that the set of algebraic numbers is uncountable?
  - seanhunter 1y ago
    I am probably wrong