6 ms·

> 0.5 is a real number Yes, and it is one of the few we have no problem with (7 is an other example of this class): The rationals (or the algebraic). Unfortuna

by drvd 8y ago

> 0.5 is a real number Yes, and it is one of the few we have no problem with (7 is an other example of this class): The rationals (or the algebraic). Unfortunately the vast majority of the reals are not that catchy. E.g. we still do not know how many reals there are. The reals are a beast if watched from a fundamental perspective.

OskarS 8y ago
> E.g. we still do not know how many reals there are. Umm... yeah we do, the cardinality of the continuum is the same as the cardinality of the power set of natural numbers. We've known this for 140 years. Reading between the lines, I think the author is really talking about the "non-computable numbers" (i.e. those real numbers who can't be calculated to an arbitrary precision by any Turing machine), but if that's what the author is referring to, he should just say "non-computable numbers", not "real numbers", which is a much broader class.
- danharaj 8y ago
  What is the cardinality of the powerset of the naturals?
  - OskarS 8y ago
    It's usually referred to as "c". I'm not sure what you're asking? It's like saying "3 is the number that follows 2", and then asking "yeah, but what really IS the number 3, man?". The answer is: it's the cardinality of the sets which can be in a bijection with the power set of the natural numbers. If you're referring to the continuum hypothesis, that is a distinct question from "what is the cardinality of the real numbers". Also: a solved one, the contiuum hypothesis is independent of ZFC, so saying "we don't even know what the cardinality is" is still wrong.
    - sykh 8y ago
      The cardinality of the continuum is a cardinal number. It’s one of the alephs. It is not known which aleph it is. So it’s not known what the cardinality of the powerset of the naturals is. It’s just known that it is the same cardinal number of the reals. Basically, we have two jars of marbles that contain the same number of marbles but it’s not known how many marbles that is.
      - jordigh 8y ago
        The continuum hypothesis being independent just means that it's an additional rule you can add or remove from the game you are trying to play. It doesn't mean we are lacking in knowledge and that if we were to work harder we would solve this problem. We do know which aleph c is: it's aleph_1 with CH and some other aleph without CH. Just take your pick which version you like better. It's not like we don't know which one is the true model of military combat: chess or checkers. They're just two different games with two different rule sets, and you get to pick which one you like to play more.
        OskarS 8y ago
        Exactly right.
        sykh 8y ago
        The set theory that most working mathematicians deal with is ZFC. In ZFC it is not known what cardinal the continuum is. Hence the statement that I was responding to is incorrect. The person I responded to said that they do know how many reals there are. The cardinality of the reals is called c. It is known to the be the same as the cardinality of the power set of the naturals. It is not known, in ZFC, which aleph this is. We just know that it is the same as the size of another set. If you want to add an axiom and say that c is aleph1 then you are free to do so. But if you don't have this axiom then you don't know which aleph it is. So in what sense can you say that you know how many reals there are? You only know it if you add an axiom that says, "It is aleph1." If I have a jar of pennies and I know it has the same number of pennies as the number of quarters in another jar that I have, does this mean I know how many pennies are in the jar?
      - 8y ago
    - danharaj 8y ago
      This is an extremely unusual conception of epistemology, even for mathematics. This knowledge that is contingent on axioms isn't at all convincing. At the beginning of the 20th century, this was a raging debate. It wasn't quite resolved, it just didn't quite matter to practicing mathematicians so it kind of faded into the background. There is a massive gap between 3 and the cardinality of the continuum. 3 is directly examinable. If I count some collection of objects, my fingers say, I will immediately grasp 3. On the other hand, the set of real numbers is a highly pathological, abstract concept. Everything we know about the reals suffers from two severe deficiencies: One, it depends on infinitary axioms which presuppose facts about the phenomenon we would like to investigate. Two, even what we can infer from such axioms is always indirect evidence. That's not surprising: All reasoning about infinity is indirect. In certain mathematical settings it is even true that the Cauchy sequence definition of real numbers and the Dedekind cut definition do not agree. At the very least, the reals isn't even a thing: there are many reals. That you say that "we know the cardinality of the reals" because we know CH is independent of ZFC is... preposterous to say the least. "If you assume you know the cardinality of the reals, then you know the cardinality of the reals; therefore we know the cardinality of the reals" is basically what such axiomatic acrobatics boils down to. Axioms are not knowledge.
      - no_identd 8y ago
        Here, another interesting construction of the real numbers: https://arxiv.org/abs/math/0405454 https://arxiv.org/abs/math/0405454 - The Eudoxus Real Numbers https://ncatlab.org/nlab/show/Eudoxus+real+number https://ncatlab.org/nlab/show/Eudoxus+real+number https://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Construction_from_Z_.28Eudoxus_reals.29 https://en.wikipedia.org/wiki/Construction_of_the_real_numbe... (Surprisingly enough, ncat & Wikipedia complement each other here in their explanations of it.)
      - gowld 8y ago
        Getting back to the OP: in a very real sense, the set of Real numbers does not exist. Therefore, it is impossible to know its size via experience. The only possible way is to derive it from chosen axioms. Note that even your conception of 3 is reliant on axioms. How do you "know" that 3 ducks is the same 3 as 3 fingers? Only via axioms.
  - wierd0 8y ago
    > What is the cardinality of the powerset of the naturals? That question is better than almost every other question!
- btilly 8y ago
  You overstate "knowledge". We project back currently accepted axioms, and find that Cantor's proof works. And therefore it was known, because the proof was known. However Cantor's proof is not so cut and dry, nor was it so unarguable at the time. It was based on set theory, and it was not clear to people at the time that set theory actually worked. Indeed, in 1901 Bertrand Russell came up with "the set of all sets that do not contain themselves" and came to a contradiction. One of the proposed resolutions was to find a better set of axioms, which lead us to ZF and later to ZFC. This is the path that mathematics took. Another was to question what words like "exists" and "truth" mean. In particular, does it make sense to talk about the existence of something we cannot construct? To talk about the truth of a statement that we have neither proof nor disproof of? This path leads to constructivism, and in constructivism Cantor's "proof" isn't a proof at all! As it turns out, there are philosophical reasons to prefer constructivism, but mathematics is easier to do within formalism. After mathematicians gained enough experience with and trust for ZF, they went with convenience. But there are plenty of mathematicians historically, and even a few remaining today, who think that the entire tower of cardinalities from classical set theory is formal nonsense meaning nothing. And there is no logical flaw in their views.
  - chriswarbo 8y ago
    > in constructivism Cantor's "proof" isn't a proof at all! Are you sure about that, and could you point me to a reference if so? It was my understanding that Cantor's proof works perfectly well in a constructive setting. For example, we can represent real numbers in a constructive way as functions from a natural number to a digit, where we interpret f(N) as giving us the Nth decimal place. We can represent an infinite list in the same way: taking a natural number and giving the value at that list position (a real number is hence an infinite list of digits). Hence we can construct a function F which takes in a list of real numbers (a function mapping natural numbers to functions-mapping-natural-numbers-to-digits) and gives out a real number (a function mapping natural numbers to digits). Given a number N, this function looks up the Nth digit of the Nth number in the list, and returns a different digit (e.g. one more, modulo 10). Formally, in some Agda-like notation, it would be something like: Digit = Member of {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} inc : Digit -> Digit inc(0) = 1 ... inc(9) = 0 List(t) = Natural -> t Real = List(Digit) Real = Natural -> Digit F : List(Real) -> Real F : (Natural -> Real) -> Real F : (Natural -> (Natural -> Digit)) -> (Natural -> Digit) F(list)(n) = inc(list(n)(n)) Cantor's proof is then a function which takes a list L and returns a proof that F(L) doesn't occur in L. We can represent non-occurrence using a function taking a natural number N and returning a proof that the Nth element of L differs from F(L) (this is a list of proofs!). We can represent proofs that two numbers differ by using a natural number, which gives (one of) the decimal places at which those numbers have different digits. We know from the definition of F that F(L) will differ from the Nth value in the list, and more specifically that it will differ at the Nth decimal place. It's easy to prove that distinct digits are different, since the set of digits is finite (although the exact encoding depends on the logical system being used). Hence we just need to show that `inc(N) != N`, and use that as proof that `F(L)(N) != L(N)(N)`, and hence `F(L)` doesn't occur in `L`: -- Proof that x != y, however you want to represent that (e.g. 'Equal x y -> Empty') Differ(t) : (x : t) -> (y : t) -> Set of proofs that x != y incDiffers : (d : Digit) -> Differ(Digit)(d, inc(d)) incDiffers(0) = 0 != 1 ... incDiffers(9) = 9 != 0 -- Proof that two reals differ, which is a natural and a proof that they differ -- at that digit RealsDiffer(x, y) = {n : Natural | Differ(Digit)(x(n), y(n))} -- Proof that F(L) differs from everything in L. This is because the Nth element of L -- differs from F(L) at (at least) the Nth decimal place. cantor : (L : List(Real)) -> (n : Natural) -> RealsDiffer(L(n), F(L)) cantor(L)(n) = (n, incDiffers(L(n)(n)))
- xamuel 8y ago
  Presumably drvd is referring to the independence of the Continuum Hypothesis: we don't know |R| in the sense that we don't know for which ordinal alpha is |R| the alpha'th infinite cardinal. The Continuum Hypothesis is the claim that |R| is the first cardinal after |N|. It's well known the Continuum Hypothesis is independent of ZFC. That means for the question of "how many reals there are" (at least if that question is read in the formal sense), the answer is "it depends: for different models of ZFC, there are different answers".

by drvd 8y ago

OskarS 8y ago

> E.g. we still do not know how many reals there are. Umm... yeah we do, the cardinality of the continuum is the same as the cardinality of the power set of natural numbers. We've known this for 140 years. Reading between the lines, I think the author is really talking about the "non-computable numbers" (i.e. those real numbers who can't be calculated to an arbitrary precision by any Turing machine), but if that's what the author is referring to, he should just say "non-computable numbers", not "real numbers", which is a much broader class.

danharaj 8y ago
What is the cardinality of the powerset of the naturals?
- OskarS 8y ago
  It's usually referred to as "c". I'm not sure what you're asking? It's like saying "3 is the number that follows 2", and then asking "yeah, but what really IS the number 3, man?". The answer is: it's the cardinality of the sets which can be in a bijection with the power set of the natural numbers. If you're referring to the continuum hypothesis, that is a distinct question from "what is the cardinality of the real numbers". Also: a solved one, the contiuum hypothesis is independent of ZFC, so saying "we don't even know what the cardinality is" is still wrong.
  - sykh 8y ago
    The cardinality of the continuum is a cardinal number. It’s one of the alephs. It is not known which aleph it is. So it’s not known what the cardinality of the powerset of the naturals is. It’s just known that it is the same cardinal number of the reals. Basically, we have two jars of marbles that contain the same number of marbles but it’s not known how many marbles that is.
    - jordigh 8y ago
      The continuum hypothesis being independent just means that it's an additional rule you can add or remove from the game you are trying to play. It doesn't mean we are lacking in knowledge and that if we were to work harder we would solve this problem. We do know which aleph c is: it's aleph_1 with CH and some other aleph without CH. Just take your pick which version you like better. It's not like we don't know which one is the true model of military combat: chess or checkers. They're just two different games with two different rule sets, and you get to pick which one you like to play more.
      - OskarS 8y ago
        Exactly right.
      - sykh 8y ago
        The set theory that most working mathematicians deal with is ZFC. In ZFC it is not known what cardinal the continuum is. Hence the statement that I was responding to is incorrect. The person I responded to said that they do know how many reals there are. The cardinality of the reals is called c. It is known to the be the same as the cardinality of the power set of the naturals. It is not known, in ZFC, which aleph this is. We just know that it is the same as the size of another set. If you want to add an axiom and say that c is aleph1 then you are free to do so. But if you don't have this axiom then you don't know which aleph it is. So in what sense can you say that you know how many reals there are? You only know it if you add an axiom that says, "It is aleph1." If I have a jar of pennies and I know it has the same number of pennies as the number of quarters in another jar that I have, does this mean I know how many pennies are in the jar?
    - 8y ago
  - danharaj 8y ago
    This is an extremely unusual conception of epistemology, even for mathematics. This knowledge that is contingent on axioms isn't at all convincing. At the beginning of the 20th century, this was a raging debate. It wasn't quite resolved, it just didn't quite matter to practicing mathematicians so it kind of faded into the background. There is a massive gap between 3 and the cardinality of the continuum. 3 is directly examinable. If I count some collection of objects, my fingers say, I will immediately grasp 3. On the other hand, the set of real numbers is a highly pathological, abstract concept. Everything we know about the reals suffers from two severe deficiencies: One, it depends on infinitary axioms which presuppose facts about the phenomenon we would like to investigate. Two, even what we can infer from such axioms is always indirect evidence. That's not surprising: All reasoning about infinity is indirect. In certain mathematical settings it is even true that the Cauchy sequence definition of real numbers and the Dedekind cut definition do not agree. At the very least, the reals isn't even a thing: there are many reals. That you say that "we know the cardinality of the reals" because we know CH is independent of ZFC is... preposterous to say the least. "If you assume you know the cardinality of the reals, then you know the cardinality of the reals; therefore we know the cardinality of the reals" is basically what such axiomatic acrobatics boils down to. Axioms are not knowledge.
    - no_identd 8y ago
      Here, another interesting construction of the real numbers: https://arxiv.org/abs/math/0405454 https://arxiv.org/abs/math/0405454 - The Eudoxus Real Numbers https://ncatlab.org/nlab/show/Eudoxus+real+number https://ncatlab.org/nlab/show/Eudoxus+real+number https://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Construction_from_Z_.28Eudoxus_reals.29 https://en.wikipedia.org/wiki/Construction_of_the_real_numbe... (Surprisingly enough, ncat & Wikipedia complement each other here in their explanations of it.)
    - gowld 8y ago
      Getting back to the OP: in a very real sense, the set of Real numbers does not exist. Therefore, it is impossible to know its size via experience. The only possible way is to derive it from chosen axioms. Note that even your conception of 3 is reliant on axioms. How do you "know" that 3 ducks is the same 3 as 3 fingers? Only via axioms.
- wierd0 8y ago
  > What is the cardinality of the powerset of the naturals? That question is better than almost every other question!
btilly 8y ago
You overstate "knowledge". We project back currently accepted axioms, and find that Cantor's proof works. And therefore it was known, because the proof was known. However Cantor's proof is not so cut and dry, nor was it so unarguable at the time. It was based on set theory, and it was not clear to people at the time that set theory actually worked. Indeed, in 1901 Bertrand Russell came up with "the set of all sets that do not contain themselves" and came to a contradiction. One of the proposed resolutions was to find a better set of axioms, which lead us to ZF and later to ZFC. This is the path that mathematics took. Another was to question what words like "exists" and "truth" mean. In particular, does it make sense to talk about the existence of something we cannot construct? To talk about the truth of a statement that we have neither proof nor disproof of? This path leads to constructivism, and in constructivism Cantor's "proof" isn't a proof at all! As it turns out, there are philosophical reasons to prefer constructivism, but mathematics is easier to do within formalism. After mathematicians gained enough experience with and trust for ZF, they went with convenience. But there are plenty of mathematicians historically, and even a few remaining today, who think that the entire tower of cardinalities from classical set theory is formal nonsense meaning nothing. And there is no logical flaw in their views.
- chriswarbo 8y ago
  > in constructivism Cantor's "proof" isn't a proof at all! Are you sure about that, and could you point me to a reference if so? It was my understanding that Cantor's proof works perfectly well in a constructive setting. For example, we can represent real numbers in a constructive way as functions from a natural number to a digit, where we interpret f(N) as giving us the Nth decimal place. We can represent an infinite list in the same way: taking a natural number and giving the value at that list position (a real number is hence an infinite list of digits). Hence we can construct a function F which takes in a list of real numbers (a function mapping natural numbers to functions-mapping-natural-numbers-to-digits) and gives out a real number (a function mapping natural numbers to digits). Given a number N, this function looks up the Nth digit of the Nth number in the list, and returns a different digit (e.g. one more, modulo 10). Formally, in some Agda-like notation, it would be something like: Digit = Member of {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} inc : Digit -> Digit inc(0) = 1 ... inc(9) = 0 List(t) = Natural -> t Real = List(Digit) Real = Natural -> Digit F : List(Real) -> Real F : (Natural -> Real) -> Real F : (Natural -> (Natural -> Digit)) -> (Natural -> Digit) F(list)(n) = inc(list(n)(n)) Cantor's proof is then a function which takes a list L and returns a proof that F(L) doesn't occur in L. We can represent non-occurrence using a function taking a natural number N and returning a proof that the Nth element of L differs from F(L) (this is a list of proofs!). We can represent proofs that two numbers differ by using a natural number, which gives (one of) the decimal places at which those numbers have different digits. We know from the definition of F that F(L) will differ from the Nth value in the list, and more specifically that it will differ at the Nth decimal place. It's easy to prove that distinct digits are different, since the set of digits is finite (although the exact encoding depends on the logical system being used). Hence we just need to show that `inc(N) != N`, and use that as proof that `F(L)(N) != L(N)(N)`, and hence `F(L)` doesn't occur in `L`: -- Proof that x != y, however you want to represent that (e.g. 'Equal x y -> Empty') Differ(t) : (x : t) -> (y : t) -> Set of proofs that x != y incDiffers : (d : Digit) -> Differ(Digit)(d, inc(d)) incDiffers(0) = 0 != 1 ... incDiffers(9) = 9 != 0 -- Proof that two reals differ, which is a natural and a proof that they differ -- at that digit RealsDiffer(x, y) = {n : Natural | Differ(Digit)(x(n), y(n))} -- Proof that F(L) differs from everything in L. This is because the Nth element of L -- differs from F(L) at (at least) the Nth decimal place. cantor : (L : List(Real)) -> (n : Natural) -> RealsDiffer(L(n), F(L)) cantor(L)(n) = (n, incDiffers(L(n)(n)))
xamuel 8y ago
Presumably drvd is referring to the independence of the Continuum Hypothesis: we don't know |R| in the sense that we don't know for which ordinal alpha is |R| the alpha'th infinite cardinal. The Continuum Hypothesis is the claim that |R| is the first cardinal after |N|. It's well known the Continuum Hypothesis is independent of ZFC. That means for the question of "how many reals there are" (at least if that question is read in the formal sense), the answer is "it depends: for different models of ZFC, there are different answers".