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As a kid, I asked my mom if all numbers have a name, she said yes, and then I asked the obvious follow-up question: If there are infinite numbers, and all numbe

by sandov 8y ago

As a kid, I asked my mom if all numbers have a name, she said yes, and then I asked the obvious follow-up question: If there are infinite numbers, and all numbers have a name, then there must exist a number called door, another one named airplane, because at some point you would run out of numbery names, I don't remember what she said, but she didn't answer the question. That question was unresolved for me until recently, when I revisited the question and thought that at some point you could start naming them a, aa, aaa, aaaa, etc instead of seventen hexadecillions five hundred and sixty octillions ... or something like that. I would love to remember all the other questions I had.

jpulgarin 8y ago
If you're referring to real numbers, then virtually no real numbers have a name. If all real numbers had a name then you could order them alphabetically and put them in one-to-one correspondence with the natural numbers, which we know is impossible: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
- cyborgx7 8y ago
  That is something I hit uppon recently, when I tried to find a binary representation that could concievably represent all real numbers, given enough bits of data. I don't think I've ever seen such a practical application of countability and cardinality. Really interesting where this stuff manifests in the real world.
- wwweston 8y ago
  Does this apply if names are infinitely long?
  - cyborgx7 8y ago
    No, then the names could just be the actual decimal representation of the number. But strings are in genereal considered to be finite.
  - shoo 8y ago
    [edit: for clarity i'm not going to use the term "name"] the proof crucially relies upon each real number being represented as an infinitely long (perhaps ending in an infinite sequence of repeated digits for less interesting numbers) sequence of digits. i.e. the representation that Cantor uses for reals is infinite sequences -- effectively representing each real number infinitely long (in the countable sense) strings. Cantor represents each real number as an infinite sequence of digits. apart from technical details`+` you can think of this as the base 10 or base 2 expansion of the number. for simplicity, you need only consider counting the real numbers between 0 and 1. there are plenty enough of those. each such number can be addressed in base 2 as some infinite sequence of 1s and 0s after the binary point. Cantor's argument shows that if you try to enumerate all such numbers (i.e. count them) by their series expansion, then you can generate a new number that doesnt apppear in the enumerated list, which is a proof by contradiction, refuting the assumption that you could enumerate them all in the first place. `+` : technical details include that some numbers have non-unique representations as infinite sequences of digits. For example, in base 10, 1 can be represented as 1.000... (where the 0s keep going) or 0.999... (where the 9s keep going). if this irritates you, let z = 0.999... then 10 * z = 9.999... then 10 * z - z = 9 then 9 * z = 9 so z = 1
- shoo 8y ago
  to continue the discussion, there are many kinds of infinities, and some are larger than others. one of the smaller kinds of infinities is "countable". we say a collection of countably infinite things is countable if, intuitively, we can count them (as the parent says, put them in 1:1 correspondence with the natural numbers 1, 2, 3, ....). there are only a countably infinite number of rational numbers, since each rational number has the form p / q, where p and q are (perhaps negative) integers. if we made a large 2d grid of all integer grid points, we could regard each rational number p/q as a grid point (p, q). Then we can count the grid points by starting at the origin of the grid (0, 0) * and spiraling outwards. This will eventually count every grid point, so this gives us a way to count all the rationals. as the parent post says, by Cantor's diagonalisation argument we can demonstrate that we can't count the real numbers, so there are a lot more reals than rationals. it's a strictly bigger kind of infinity. even if we start inventing new notation for particular reals we care about (e.g. pi, e, pi^e, door, super(door, |^|^bat)man, ) -- whatever you like provided it is well-defined, we can only name at most countably many reals, leaving a remainder of uncountably many un-named reals. we can single out any particular real that can be well-defined, and mint a new name for it, but we can only do this for at most countably many such reals, while the bulk of the reals escape naming. * the origin (0, 0) corresponds to 0 / 0 which isn't a rational number, and some rational numbers such as 4 have multiple representations as coordinates. For example we could write 4 as 4 / 1 or 8 / 2 or 40 / 10 or -16 / -4 ... so strictly speaking by demonstrating we can count all of the 2d integer grid points shows that there are at most a countably infinite number of rationals. but since each natural number is a rational, and there are countably many rationals, we know there's at least a countably infinite number of rationals.
- jgeralnik 8y ago
  Not quite, you can't order all natural numbers alphabetically either but that doesn't mean you can't count them. A better way to count based on names would be to first count all single letter names alphabetically, then all two letter names alphabetically, etc.
  - enedil 8y ago
    Yes, you can. Order is just a relation, i.e. a set that contains pairs of elements. So for example, let X be the relation <. X is a subset of NxN, the cartesian product of N. We say that n<m if the pair (n,m) is in X. Moreover, we say that (n,n+1) is in X, and whenever (n,m) in X and (m,k) in X, then (n,k) in X. This constitues order on naturals.
    - jgeralnik 8y ago
      True, but the GP claimed that if we can order alphabetically we can create a one to one correspondence with natural numbers. Obviously we can order reals (using <), I interpreted the original post as referring not to your definition but to "order and count"
cyborgx7 8y ago
The difference between "infinite" and "all possible" seems to be really unintuitive. The example I got to is that there are infinite even numbers, but not all natural numbers are even. This also maniests itself when people talk about infinite universes. "If there are infinite universes, there has to be one in which X". Not necessarialy. Maybe only even universes exist and universes in which X is the case might all be odd universes.
- hrnnnnnn 8y ago
  I heard a really good interpretation of this recently. In all these infinite universes, surely there's one where there's an ice cube in the middle of the sun, since that's one possible configuration of matter. But no! There's no story that ends with an ice-cube in the middle of the sun. There's no sequence of events that could have resulted in that so it's not in the space of possible universes.
  - Confusion 8y ago
    Why do you believe there is no sequence of events that could end with an ice cube in the middle of a sun? One scenario I would not rule out is a large icy comet moving at large speed straight for a sun. Is there no combination of size and speed for which that would result in a small ice cube in the centre of a sun?
    - skriticos2 8y ago
      Hm, it would probably not be cube shaped though. But I agree, we could even calculate that one. It would certainly be an interesting spectacle. Going back to the sequence of events argument, I think that implies that all universes agree on similar rules like ours (Physics), as we could only deduce this kind of argument with this assumption. Maybe there is a universe where ice is scorching hot and all sun cores consist of it, and as matter cools it evaporates? Maybe there is even a universe where gravity and orbital mechanics work in a way to produce cube shaped stellar bodies?
  - Razengan 8y ago
SideburnsOfDoom 8y ago
> If there are infinite numbers, and all numbers have a name, then there must exist a number called door, another one named airplane, because at some point you would run out of numbery names, I don't remember what she said, but she didn't answer the question. Ah, this is the philosophical distinction between every possible thing, and every conceivable thing. The set of possible number names may be very large or infinite, but it is assembled from a small set of symbols like "one", "two" "seventeen" "million", "and", "times" and "three" "to the power of", etc. A number called "door" is conceivable (like superman) but it is not a possible under the rules of maths (or physics in superman's case) because "door" is not in that set of symbols. The set of the possible is a subset of the conceivable. Both sets might be infinite.
- azeirah 8y ago
  > A number called "door" is conceivable (like superman) but it is not a possible under the rules of maths (or physics in superman's case) because "door" is not in that set of symbols. Certainly you are able to represent "door" in a base-64 numbering system? (or base-36, or even base-28)
  - SideburnsOfDoom 8y ago
    Ah, but "a base-64 numbering system" is a different set of rules ;) And it's rules in which "door" is a "numbery name". And "Достоевский" is not. You could still ask "If there are infinite base-64 numbers, and all base-64 numbers have a name, then must there exist a number called "Достоевский" ?" and the same argument against would apply.
- n4r9 8y ago
  I think the distinction is really between "A given infinite set" and "every conceivable thing". It's a common fallacy to equate one infinity with any other infinity, and therefore any given infinity must contain all things. This fallacy is very neatly encapsulated by the phrase >If there are infinite numbers, and all numbers have a name, then there must exist a number called door

by sandov 8y ago

jpulgarin 8y ago

If you're referring to real numbers, then virtually no real numbers have a name. If all real numbers had a name then you could order them alphabetically and put them in one-to-one correspondence with the natural numbers, which we know is impossible: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

cyborgx7 8y ago
That is something I hit uppon recently, when I tried to find a binary representation that could concievably represent all real numbers, given enough bits of data. I don't think I've ever seen such a practical application of countability and cardinality. Really interesting where this stuff manifests in the real world.
wwweston 8y ago
Does this apply if names are infinitely long?
- cyborgx7 8y ago
  No, then the names could just be the actual decimal representation of the number. But strings are in genereal considered to be finite.
- shoo 8y ago
  [edit: for clarity i'm not going to use the term "name"] the proof crucially relies upon each real number being represented as an infinitely long (perhaps ending in an infinite sequence of repeated digits for less interesting numbers) sequence of digits. i.e. the representation that Cantor uses for reals is infinite sequences -- effectively representing each real number infinitely long (in the countable sense) strings. Cantor represents each real number as an infinite sequence of digits. apart from technical details`+` you can think of this as the base 10 or base 2 expansion of the number. for simplicity, you need only consider counting the real numbers between 0 and 1. there are plenty enough of those. each such number can be addressed in base 2 as some infinite sequence of 1s and 0s after the binary point. Cantor's argument shows that if you try to enumerate all such numbers (i.e. count them) by their series expansion, then you can generate a new number that doesnt apppear in the enumerated list, which is a proof by contradiction, refuting the assumption that you could enumerate them all in the first place. `+` : technical details include that some numbers have non-unique representations as infinite sequences of digits. For example, in base 10, 1 can be represented as 1.000... (where the 0s keep going) or 0.999... (where the 9s keep going). if this irritates you, let z = 0.999... then 10 * z = 9.999... then 10 * z - z = 9 then 9 * z = 9 so z = 1
shoo 8y ago
to continue the discussion, there are many kinds of infinities, and some are larger than others. one of the smaller kinds of infinities is "countable". we say a collection of countably infinite things is countable if, intuitively, we can count them (as the parent says, put them in 1:1 correspondence with the natural numbers 1, 2, 3, ....). there are only a countably infinite number of rational numbers, since each rational number has the form p / q, where p and q are (perhaps negative) integers. if we made a large 2d grid of all integer grid points, we could regard each rational number p/q as a grid point (p, q). Then we can count the grid points by starting at the origin of the grid (0, 0) * and spiraling outwards. This will eventually count every grid point, so this gives us a way to count all the rationals. as the parent post says, by Cantor's diagonalisation argument we can demonstrate that we can't count the real numbers, so there are a lot more reals than rationals. it's a strictly bigger kind of infinity. even if we start inventing new notation for particular reals we care about (e.g. pi, e, pi^e, door, super(door, |^|^bat)man, ) -- whatever you like provided it is well-defined, we can only name at most countably many reals, leaving a remainder of uncountably many un-named reals. we can single out any particular real that can be well-defined, and mint a new name for it, but we can only do this for at most countably many such reals, while the bulk of the reals escape naming. * the origin (0, 0) corresponds to 0 / 0 which isn't a rational number, and some rational numbers such as 4 have multiple representations as coordinates. For example we could write 4 as 4 / 1 or 8 / 2 or 40 / 10 or -16 / -4 ... so strictly speaking by demonstrating we can count all of the 2d integer grid points shows that there are at most a countably infinite number of rationals. but since each natural number is a rational, and there are countably many rationals, we know there's at least a countably infinite number of rationals.
jgeralnik 8y ago
Not quite, you can't order all natural numbers alphabetically either but that doesn't mean you can't count them. A better way to count based on names would be to first count all single letter names alphabetically, then all two letter names alphabetically, etc.
- enedil 8y ago
  Yes, you can. Order is just a relation, i.e. a set that contains pairs of elements. So for example, let X be the relation <. X is a subset of NxN, the cartesian product of N. We say that n<m if the pair (n,m) is in X. Moreover, we say that (n,n+1) is in X, and whenever (n,m) in X and (m,k) in X, then (n,k) in X. This constitues order on naturals.
  - jgeralnik 8y ago
    True, but the GP claimed that if we can order alphabetically we can create a one to one correspondence with natural numbers. Obviously we can order reals (using <), I interpreted the original post as referring not to your definition but to "order and count"

cyborgx7 8y ago

The difference between "infinite" and "all possible" seems to be really unintuitive. The example I got to is that there are infinite even numbers, but not all natural numbers are even. This also maniests itself when people talk about infinite universes. "If there are infinite universes, there has to be one in which X". Not necessarialy. Maybe only even universes exist and universes in which X is the case might all be odd universes.

hrnnnnnn 8y ago
I heard a really good interpretation of this recently. In all these infinite universes, surely there's one where there's an ice cube in the middle of the sun, since that's one possible configuration of matter. But no! There's no story that ends with an ice-cube in the middle of the sun. There's no sequence of events that could have resulted in that so it's not in the space of possible universes.
- Confusion 8y ago
  Why do you believe there is no sequence of events that could end with an ice cube in the middle of a sun? One scenario I would not rule out is a large icy comet moving at large speed straight for a sun. Is there no combination of size and speed for which that would result in a small ice cube in the centre of a sun?
  - skriticos2 8y ago
    Hm, it would probably not be cube shaped though. But I agree, we could even calculate that one. It would certainly be an interesting spectacle. Going back to the sequence of events argument, I think that implies that all universes agree on similar rules like ours (Physics), as we could only deduce this kind of argument with this assumption. Maybe there is a universe where ice is scorching hot and all sun cores consist of it, and as matter cools it evaporates? Maybe there is even a universe where gravity and orbital mechanics work in a way to produce cube shaped stellar bodies?
- Razengan 8y ago

SideburnsOfDoom 8y ago

> If there are infinite numbers, and all numbers have a name, then there must exist a number called door, another one named airplane, because at some point you would run out of numbery names, I don't remember what she said, but she didn't answer the question. Ah, this is the philosophical distinction between every possible thing, and every conceivable thing. The set of possible number names may be very large or infinite, but it is assembled from a small set of symbols like "one", "two" "seventeen" "million", "and", "times" and "three" "to the power of", etc. A number called "door" is conceivable (like superman) but it is not a possible under the rules of maths (or physics in superman's case) because "door" is not in that set of symbols. The set of the possible is a subset of the conceivable. Both sets might be infinite.

azeirah 8y ago
> A number called "door" is conceivable (like superman) but it is not a possible under the rules of maths (or physics in superman's case) because "door" is not in that set of symbols. Certainly you are able to represent "door" in a base-64 numbering system? (or base-36, or even base-28)
- SideburnsOfDoom 8y ago
  Ah, but "a base-64 numbering system" is a different set of rules ;) And it's rules in which "door" is a "numbery name". And "Достоевский" is not. You could still ask "If there are infinite base-64 numbers, and all base-64 numbers have a name, then must there exist a number called "Достоевский" ?" and the same argument against would apply.
n4r9 8y ago
I think the distinction is really between "A given infinite set" and "every conceivable thing". It's a common fallacy to equate one infinity with any other infinity, and therefore any given infinity must contain all things. This fallacy is very neatly encapsulated by the phrase >If there are infinite numbers, and all numbers have a name, then there must exist a number called door