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> All real numbers, and all numbers of any other variety, can be written with a finite number of symbols. That's what it means to give something a name. Counte

> All real numbers, and all numbers of any other variety, can be written with a finite number of symbols. That's what it means to give something a name. Counter-intuitively, this is not true. The vast, vast majority of real numbers cannot be named, not even in principle. Their definitions would have to be infinitely long. Or to put it another way, no matter how close two named numbers are, there is an infinite number of reals in between them. If you say, okay, sure, but some of those might be named, then pick the two closest and then there is still an infinite number of other reals in between those two! Another way to look at it is that the amount of information (measured in bits) between any two real numbers is literally infinite. If the reals were represented with binary digits, then a sequential subset of them would have a common finite prefix, and then all possible infinite bit strings would be the suffixes!

thaumasiotes 4y ago
> The vast, vast majority of real numbers cannot be named, not even in principle. Their definitions would have to be infinitely long. So what? That doesn't stop them from being named.
- plopilop 4y ago
  Usually definitions are considered to be finite. Otherwise yes you can just name a number by enumerating its decimals but what's the point. Edit : actually does not even work, courtesy of the Chaitin's numbers
- Smaug123 4y ago
  You're using "being named" in a very unusual sense. Most people would consider you not to have named something if you're literally never allowed to stop speaking (or writing) the name. Most reals "have infinitely long names" under your definition, and there is no finite time at which you've distinguished between reals with the same initial segment of "name", so really what's the use of the naming scheme at all? (For a more compelling argument, the equality of real numbers is uncomputable, so it must be impossible to name them. If we could name them usefully, then we could determine equality by asking which ones had different names.)
  - thaumasiotes 4y ago
    > You're using "being named" in a very unusual sense. No, I'm not. > Most people would consider you not to have named something if you're literally never allowed to stop speaking (or writing) the name. What's the connection? Names do not completely describe their referent. They're names. I know someone named Ko. Given that name, what can you tell me about Ko?
    - jiggawatts 4y ago
      Naming a number in mathematics has different rules to assigning names to objects in the physical world. You can "name" a number such as: "a positive number, that when multiplied by itself, the result is 2". This is a finite statement that requires only a handful of 'bits' to represent, but it exactly and uniquely identifies the square root of two. If written with decimal digits, then the square root of two would require an infinite number of digits, but it can be defined ('named') with a finite number of bits. It turns out that this is a rare property for real numbers. The vast majority do not have a shorthand name like this, only the infinitely long description exists.
      - riversflow 4y ago
        What does naming have to do with AC though? This seems completely tangential to the axiom. It’s about the nature of sets, not the semantics of how you make the selection and reference it. AC says: if there is a set of non-empty sets y with length x, I can construct a new set z of length x by taking a member from each set in y. How does being able to name the constituents z or any member of y matter to the logic of the axiom?
        Smaug123 4y ago
        This is all in response to a comment which basically said "why do people think AC is obvious - if you can't even name all elements of the set, why would you expect to be able to choose them". (Which is handwavy, of course, and leads to the wrong intuition about uncountable-but-well-ordered sets, and depending on how you look at it it may even lead you to conclude that you can't ever choose even a single element from an uncountable set.)
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thaumasiotes 4y ago

> The vast, vast majority of real numbers cannot be named, not even in principle. Their definitions would have to be infinitely long. So what? That doesn't stop them from being named.

plopilop 4y ago
Usually definitions are considered to be finite. Otherwise yes you can just name a number by enumerating its decimals but what's the point. Edit : actually does not even work, courtesy of the Chaitin's numbers
Smaug123 4y ago
You're using "being named" in a very unusual sense. Most people would consider you not to have named something if you're literally never allowed to stop speaking (or writing) the name. Most reals "have infinitely long names" under your definition, and there is no finite time at which you've distinguished between reals with the same initial segment of "name", so really what's the use of the naming scheme at all? (For a more compelling argument, the equality of real numbers is uncomputable, so it must be impossible to name them. If we could name them usefully, then we could determine equality by asking which ones had different names.)
- thaumasiotes 4y ago
  > You're using "being named" in a very unusual sense. No, I'm not. > Most people would consider you not to have named something if you're literally never allowed to stop speaking (or writing) the name. What's the connection? Names do not completely describe their referent. They're names. I know someone named Ko. Given that name, what can you tell me about Ko?
  - jiggawatts 4y ago
    Naming a number in mathematics has different rules to assigning names to objects in the physical world. You can "name" a number such as: "a positive number, that when multiplied by itself, the result is 2". This is a finite statement that requires only a handful of 'bits' to represent, but it exactly and uniquely identifies the square root of two. If written with decimal digits, then the square root of two would require an infinite number of digits, but it can be defined ('named') with a finite number of bits. It turns out that this is a rare property for real numbers. The vast majority do not have a shorthand name like this, only the infinitely long description exists.
    - riversflow 4y ago
      What does naming have to do with AC though? This seems completely tangential to the axiom. It’s about the nature of sets, not the semantics of how you make the selection and reference it. AC says: if there is a set of non-empty sets y with length x, I can construct a new set z of length x by taking a member from each set in y. How does being able to name the constituents z or any member of y matter to the logic of the axiom?
      - Smaug123 4y ago
        This is all in response to a comment which basically said "why do people think AC is obvious - if you can't even name all elements of the set, why would you expect to be able to choose them". (Which is handwavy, of course, and leads to the wrong intuition about uncountable-but-well-ordered sets, and depending on how you look at it it may even lead you to conclude that you can't ever choose even a single element from an uncountable set.)
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