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Yes, but it also messes with a lot of normal intuitions. Some examples: “Because rationals are dense —between any two rationals there are infinitely many other

Yes, but it also messes with a lot of normal intuitions. Some examples: “Because rationals are dense —between any two rationals there are infinitely many other rationals—there are actually vastly more spaces between rational numbers, than rational numbers themselves. These spaces-between are the real numbers.” “Because every finite text document can be converted to UTF-8 and thus then an integer, it is only possible to describe 0% of the real numbers between 0 and 1 with text.” “Since most numbers are indescribable, there are (discontinuous) functions which have the value 2 for almost all numbers, but any number that you actually can describe and try to evaluate the function on, gives 1 and not 2.” You start to appreciate that logic itself is this Lovecraftian eldritch-horror abomination, and that we only live in the Bliss of Sanity because we live in ignorance, never staring into its depths lest the abyss stare directly back into our souls.

tomrod 2y ago
> You start to appreciate that logic itself is this Lovecraftian eldritch-horror abomination, and that we only live in the Bliss of Sanity because we live in ignorance, never staring into its depths lest the abyss stare directly back into our souls. Oh poppycock. We are the eldritch horror. We are the universe experiencing itself. Humans are space orcs, if Reddit is to be believed.
markusde 2y ago
> describe 0% of the rational numbers between 0 and 1 I think you mean irrational :)
- Edwinr95 2y ago
  No, the statement holds perfectly fine for the rationals.
  - LudwigNagasena 2y ago
    There is a 1-to-1 mapping between integers and rationals.
  - crdrost 2y ago
    The basis for this statement holds, but what the statement implies does not. That is, the numbers are a subset of the rationals, but it does not follow that we can't describe a rational with a number. In fact the rationals between [0, 1) have a well known numbering, [ 0/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, 1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 1/8, 3/8, ... ] where one increments the denominator and then goes through all numerators but keeps only numerators which have GCD 1 with the denominator (since if they share a factor they were already listed).
- Perseids 2y ago
  What you were probably thinking of is that 0% of the irrational numbers between 0 and 1 can be described by language as single entities. Or phrased differently: If you had a magic machine that could pick a random real number between 0 and 1, with 100% probability you would get a number that no finite phrase / definition / program / book could define. That is because everything we can abstractly define is part of a countable set and the set of irrational number (and real numbers) is uncountable. For that reason, quite a few mathematicians view the real numbers as a useful, but ultimately absurd set. Much more sane is the set of computable numbers, that is the set of numbers for which you can find an algorithm that computes the number to arbitrary precision. (More formal: A number x is computable if there exists a Turing machine that gets as input a natural number n, terminates on all inputs, and outputs a rational number y such that |x-y|<10^-n .) Every number you ever thought of is computable, but as a mathematician, working with the set of computable numbers is much more tedious than working with real numbers.
  - tshaddox 2y ago
    > Much more sane is the set of computable numbers, that is the set of numbers for which you can find an algorithm that computes the number to arbitrary precision. But perhaps still not as sane as one may hope. It would be very sane to be able to compute, for any two numbers, which one is larger (or whether they're equal), but sadly this is not computable for the computable numbers. > Every number you ever thought of is computable, but as a mathematician, working with the set of computable numbers is much more tedious than working with real numbers. I mean, I've thought of noncomputable reals like Chaitin constants.
    - xscott 2y ago
      > It would be very sane to be able to compute, for any two numbers, which one is > larger (or whether they're equal), but sadly this is not computable for the > computable numbers. I'd like to understand - Can you explain this? It seems like it would be easy to have a Turing machines that uses the other two Turing machines, adding one digit at a time until it finds a difference. > I mean, I've thought of noncomputable reals like Chaitin constants. Heh, but how many digits can you actually provide? Not too many. So have you really thought of the number in any meaningful sense when you barely know any of its digits? Also interesting that computer languages themselves are countable, so while it's hard to specify the digits algorithmically for the Chaitin constant of any computer language, you already know that the set of ALL Chaitin constants are countable.
staunton 2y ago
> it is only possible to describe 0% of the rational numbers between 0 and 1 with text It is possible to describe 100% of the rational numbers with text. You describe the numerator, make a space, then describe the denominator. The length of the text document depends on the number described and can be arbitrarily long.
- crdrost 2y ago
  Sorry, phone autocorrected “irational” to “rational” rather than “irrational.” fixed!
andrewla 2y ago
For what it's worth, this is only true in Cantor's horrifying paradise. In the world of the intuitionists, none of this is true. Reject Cantor and embrace Brouwer and you can once again live in a world without these horrors, and all you lose is absurd statements about things true "almost everywhere" that are never true, and crazy results like Banach-Tarski that get an impossible result by doing two impossible things to set it up.
- karmakurtisaani 2y ago
  How does Brouwer work again? Is it some kind of constructionist or finitist approach?
  - crdrost 2y ago
    Yeah, it is the standard idea of limits-of-rationals, but within the intuitionist logic that Brouwer championed. The easy semantics for intuitionist logic is that every statement is about provability: “A or B or C” is a statement that one or more of these 3 proofs has been supplied. Where this gets a little bit funky is, you can still take an open mathematical problem and still encode it into the reals: “The nth bit of this number r is 1 if n is a counterexample to the conjecture, or else 0 if not.” If for each n that problem is decidable in a finite number of steps, then this is a perfectly good intuitionistic predicate with which to define a number, and so Goldbach’s conjecture for example can be phrased as “is the Goldbach real equal to 0?” You can do that in the classical approach and Brouwer doesn't limit this too much. But, now you want to assert that “the Goldbach real number is either 0 or positive.” Because you know it is on the range [0, 1] by construction, right?! But no no no no no, if you want to stay that it is either 0 or positive, you have to furnish me with either a proof that it is 0 (solving the Goldbach conjecture in the affirmative), or a proof that it is positive (solving the Goldbach conjecture in the negative). So you have to come up with alternative ways to talk about the order on the real numbers because ordering statements are this classic example where people love to use the very law of the excluded middle that Brouwer has forbidden.
    - karmakurtisaani 2y ago

tomrod 2y ago

> You start to appreciate that logic itself is this Lovecraftian eldritch-horror abomination, and that we only live in the Bliss of Sanity because we live in ignorance, never staring into its depths lest the abyss stare directly back into our souls. Oh poppycock. We are the eldritch horror. We are the universe experiencing itself. Humans are space orcs, if Reddit is to be believed.

markusde 2y ago

> describe 0% of the rational numbers between 0 and 1 I think you mean irrational :)

Edwinr95 2y ago
No, the statement holds perfectly fine for the rationals.
- LudwigNagasena 2y ago
  There is a 1-to-1 mapping between integers and rationals.
- crdrost 2y ago
  The basis for this statement holds, but what the statement implies does not. That is, the numbers are a subset of the rationals, but it does not follow that we can't describe a rational with a number. In fact the rationals between [0, 1) have a well known numbering, [ 0/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, 1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 1/8, 3/8, ... ] where one increments the denominator and then goes through all numerators but keeps only numerators which have GCD 1 with the denominator (since if they share a factor they were already listed).
Perseids 2y ago
What you were probably thinking of is that 0% of the irrational numbers between 0 and 1 can be described by language as single entities. Or phrased differently: If you had a magic machine that could pick a random real number between 0 and 1, with 100% probability you would get a number that no finite phrase / definition / program / book could define. That is because everything we can abstractly define is part of a countable set and the set of irrational number (and real numbers) is uncountable. For that reason, quite a few mathematicians view the real numbers as a useful, but ultimately absurd set. Much more sane is the set of computable numbers, that is the set of numbers for which you can find an algorithm that computes the number to arbitrary precision. (More formal: A number x is computable if there exists a Turing machine that gets as input a natural number n, terminates on all inputs, and outputs a rational number y such that |x-y|<10^-n .) Every number you ever thought of is computable, but as a mathematician, working with the set of computable numbers is much more tedious than working with real numbers.
- tshaddox 2y ago
  > Much more sane is the set of computable numbers, that is the set of numbers for which you can find an algorithm that computes the number to arbitrary precision. But perhaps still not as sane as one may hope. It would be very sane to be able to compute, for any two numbers, which one is larger (or whether they're equal), but sadly this is not computable for the computable numbers. > Every number you ever thought of is computable, but as a mathematician, working with the set of computable numbers is much more tedious than working with real numbers. I mean, I've thought of noncomputable reals like Chaitin constants.
  - xscott 2y ago
    > It would be very sane to be able to compute, for any two numbers, which one is > larger (or whether they're equal), but sadly this is not computable for the > computable numbers. I'd like to understand - Can you explain this? It seems like it would be easy to have a Turing machines that uses the other two Turing machines, adding one digit at a time until it finds a difference. > I mean, I've thought of noncomputable reals like Chaitin constants. Heh, but how many digits can you actually provide? Not too many. So have you really thought of the number in any meaningful sense when you barely know any of its digits? Also interesting that computer languages themselves are countable, so while it's hard to specify the digits algorithmically for the Chaitin constant of any computer language, you already know that the set of ALL Chaitin constants are countable.

staunton 2y ago

> it is only possible to describe 0% of the rational numbers between 0 and 1 with text It is possible to describe 100% of the rational numbers with text. You describe the numerator, make a space, then describe the denominator. The length of the text document depends on the number described and can be arbitrarily long.

crdrost 2y ago
Sorry, phone autocorrected “irational” to “rational” rather than “irrational.” fixed!

andrewla 2y ago

For what it's worth, this is only true in Cantor's horrifying paradise. In the world of the intuitionists, none of this is true. Reject Cantor and embrace Brouwer and you can once again live in a world without these horrors, and all you lose is absurd statements about things true "almost everywhere" that are never true, and crazy results like Banach-Tarski that get an impossible result by doing two impossible things to set it up.

karmakurtisaani 2y ago
How does Brouwer work again? Is it some kind of constructionist or finitist approach?
- crdrost 2y ago
  Yeah, it is the standard idea of limits-of-rationals, but within the intuitionist logic that Brouwer championed. The easy semantics for intuitionist logic is that every statement is about provability: “A or B or C” is a statement that one or more of these 3 proofs has been supplied. Where this gets a little bit funky is, you can still take an open mathematical problem and still encode it into the reals: “The nth bit of this number r is 1 if n is a counterexample to the conjecture, or else 0 if not.” If for each n that problem is decidable in a finite number of steps, then this is a perfectly good intuitionistic predicate with which to define a number, and so Goldbach’s conjecture for example can be phrased as “is the Goldbach real equal to 0?” You can do that in the classical approach and Brouwer doesn't limit this too much. But, now you want to assert that “the Goldbach real number is either 0 or positive.” Because you know it is on the range [0, 1] by construction, right?! But no no no no no, if you want to stay that it is either 0 or positive, you have to furnish me with either a proof that it is 0 (solving the Goldbach conjecture in the affirmative), or a proof that it is positive (solving the Goldbach conjecture in the negative). So you have to come up with alternative ways to talk about the order on the real numbers because ordering statements are this classic example where people love to use the very law of the excluded middle that Brouwer has forbidden.
  - karmakurtisaani 2y ago