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Arbitrary precision is irrelevant. By it we mean arbitrary finite precision, which leaves us not only with computable numbers, but rationals. Representing ratio

Arbitrary precision is irrelevant. By it we mean arbitrary finite precision, which leaves us not only with computable numbers, but rationals. Representing rationals isn't affected by the paper's findings, because they have the same cardinality as the naturals. Arbitrary precision models reals, it doesn't actually use them.

mike00632 8y ago
>We can not obtain them to feed them as input, our real-world machines can not process them if we could Suppose we are only working with rational multiples of Pi. Then we can ONLY have finite calculations of irrational numbers and it's impossible to have a finite calculation of a rational number (except for multiplication by 0). >Representing rationals isn't affected by the paper's findings, because they have the same cardinality as the naturals. The Natural numbers are still infinite which presents its own inherent difficulty. It's not as though countability lends itself better to programming.
- roywiggins 8y ago
  All of the rationals are computable. Almost all of the irrationals aren't computable at all... so there's a big difference there.
  - mike00632 8y ago
    Yeah, and I suspect that has a lot to do with the underlying mathematical principles too. But the paper instead reduces the issue to the Continuum Hypothesis. The problem might not even have anything to do with countability but instead be fundamentally about the differences between infinite cardinals.
- throwawaymath 8y ago
  First and foremost: > The Natural numbers are still infinite which presents its own inherent difficulty. It's not as though countability lends itself better to programming Actually countability does lend itself to programming, essentially by definition. You cannot write a function to enumerate all the real numbers precisely because they're uncountable, even if you had an infinite amount of time. You can trivially do this with a "merely" countably infinite set, like the naturals or rationals. > Suppose we are only working with rational multiples of Pi. Then we can ONLY have finite calculations of irrational numbers and it's impossible to have a finite calculation of a rational number (except for multiplication by 0). It sounds like you're suggesting we can store an irrational number in a finite representation by generating it with a finite function (e.g. multiply a rational number of times). But that doesn't work. You can't represent a number N in a finite number of steps if you can't represent all factors of N in a finite number of steps. This fact follows from the field axioms for the same reason that all products of irrationals and rationals are themselves irrational. Simply put, if you can't do it for Pi, you can't do it for a rational multiple of Pi.
  - mike00632 8y ago
    >You cannot write a function to enumerate all the real numbers precisely because they're uncountable, even if you had an infinite amount of time Even with infinite amount of time your enumeration function will not generate all of the natural numbers even though the natural number are countable. You're right that we can't write a function to determine an immediate successor but we can write successor functions. >It sounds like you're suggesting we can store an irrational number in a finite representation by generating it with a finite function (e.g. multiply a rational number of times). But that doesn't work. You can't represent a number N in a finite number of steps if you can't represent all factors of N in a finite number of steps. We are getting a few concepts muddled. Storing an irrational number as a finite representation is trivially easy. For example: "pi". Alternatively I could write an algorithm to generate the decimal expansion of the square root of 2. Where it's difficult is when we talk about numbers that aren't describable in a finite number of terms, i.e. Uncomputable. There is a certain unknowable quality to these numbers and I would suggest that this isn't inherent from the cardinality of the real numbers and that we run into similar problems when trying to compute some natural numbers, like random numbers.
    - throwawaymath 8y ago
      > Even with infinite amount of time your enumeration function will not generate all of the natural numbers even though the natural number are countable. Huh? You can trivially write a function to generate all natural numbers: def increment(x): return increment(x + 1) increment(1) How do you propose doing that for the reals? > Where it's difficult is when we talk about numbers that aren't describable in a finite number of terms, i.e. Uncomputable. This is not what uncomputable means. Uncomputable means we cannot calculate a number to any precision, not that the number has an infinite decimal expansion. Pi is not uncomputable. You can’t store its entire decimal expansion in finite space but it is certainly computable - we compute it all the time. However, almost all real numbers are strictly uncomputable. Among many other proofs, this is evident from the simple fact that the set of all possible Turing machines is countable. Since the reals are uncountable, it follows that almost all reals must not be computable by any Turing machine. Likewise, there are many potential calculations involving real numbers for which a computer will output a strictly incorrect - not just imprecise - result because its real solution is uncomputable. This does not happen in the naturals or rationals.
- lmm 8y ago
  > Suppose we are only working with rational multiples of Pi. How would you start? You can't put a rational multiple of pi exactly into a computer. You can't get a rational multiple of pi exactly out of a computer. You can calculate rational numbers on the computer and scale them by a factor of pi outside the computer, if you really want, but that doesn't mean the computer is working with irrationals. > It's not as though countability lends itself better to programming. Huh? Yes it does; countable sets are still infinite but they're better-behaved than bigger sets.
  - mike00632 8y ago
    >You can't put a rational multiple of pi exactly into a computer. Of course we can: "1/2*pi" >You can't get a rational multiple of pi exactly out of a computer. Using the same limitations you seem to have in mind, we can't get one third exactly either. >...countable sets are still infinite but they're better-behaved than bigger sets. Yes, and that is surprising because we do have numerous clever applications that skirt around most of the intractability of exotic real numbers. That is why I think this paper is a bid deal, especially because it points to the cardinality of the real numbers instead of, say, their inclusion of the uncomputable numbers.
    - lmm 8y ago
      > Using the same limitations you seem to have in mind, we can't get one third exactly either. We can work with exact rationals; we can do arithmetic on them and normalize them. In particular, we can compute whether two rationals are equal (in finite time), which we can't do with reals.

mike00632 8y ago

>We can not obtain them to feed them as input, our real-world machines can not process them if we could Suppose we are only working with rational multiples of Pi. Then we can ONLY have finite calculations of irrational numbers and it's impossible to have a finite calculation of a rational number (except for multiplication by 0). >Representing rationals isn't affected by the paper's findings, because they have the same cardinality as the naturals. The Natural numbers are still infinite which presents its own inherent difficulty. It's not as though countability lends itself better to programming.

roywiggins 8y ago
All of the rationals are computable. Almost all of the irrationals aren't computable at all... so there's a big difference there.
- mike00632 8y ago
  Yeah, and I suspect that has a lot to do with the underlying mathematical principles too. But the paper instead reduces the issue to the Continuum Hypothesis. The problem might not even have anything to do with countability but instead be fundamentally about the differences between infinite cardinals.
throwawaymath 8y ago
First and foremost: > The Natural numbers are still infinite which presents its own inherent difficulty. It's not as though countability lends itself better to programming Actually countability does lend itself to programming, essentially by definition. You cannot write a function to enumerate all the real numbers precisely because they're uncountable, even if you had an infinite amount of time. You can trivially do this with a "merely" countably infinite set, like the naturals or rationals. > Suppose we are only working with rational multiples of Pi. Then we can ONLY have finite calculations of irrational numbers and it's impossible to have a finite calculation of a rational number (except for multiplication by 0). It sounds like you're suggesting we can store an irrational number in a finite representation by generating it with a finite function (e.g. multiply a rational number of times). But that doesn't work. You can't represent a number N in a finite number of steps if you can't represent all factors of N in a finite number of steps. This fact follows from the field axioms for the same reason that all products of irrationals and rationals are themselves irrational. Simply put, if you can't do it for Pi, you can't do it for a rational multiple of Pi.
- mike00632 8y ago
  >You cannot write a function to enumerate all the real numbers precisely because they're uncountable, even if you had an infinite amount of time Even with infinite amount of time your enumeration function will not generate all of the natural numbers even though the natural number are countable. You're right that we can't write a function to determine an immediate successor but we can write successor functions. >It sounds like you're suggesting we can store an irrational number in a finite representation by generating it with a finite function (e.g. multiply a rational number of times). But that doesn't work. You can't represent a number N in a finite number of steps if you can't represent all factors of N in a finite number of steps. We are getting a few concepts muddled. Storing an irrational number as a finite representation is trivially easy. For example: "pi". Alternatively I could write an algorithm to generate the decimal expansion of the square root of 2. Where it's difficult is when we talk about numbers that aren't describable in a finite number of terms, i.e. Uncomputable. There is a certain unknowable quality to these numbers and I would suggest that this isn't inherent from the cardinality of the real numbers and that we run into similar problems when trying to compute some natural numbers, like random numbers.
  - throwawaymath 8y ago
    > Even with infinite amount of time your enumeration function will not generate all of the natural numbers even though the natural number are countable. Huh? You can trivially write a function to generate all natural numbers: def increment(x): return increment(x + 1) increment(1) How do you propose doing that for the reals? > Where it's difficult is when we talk about numbers that aren't describable in a finite number of terms, i.e. Uncomputable. This is not what uncomputable means. Uncomputable means we cannot calculate a number to any precision, not that the number has an infinite decimal expansion. Pi is not uncomputable. You can’t store its entire decimal expansion in finite space but it is certainly computable - we compute it all the time. However, almost all real numbers are strictly uncomputable. Among many other proofs, this is evident from the simple fact that the set of all possible Turing machines is countable. Since the reals are uncountable, it follows that almost all reals must not be computable by any Turing machine. Likewise, there are many potential calculations involving real numbers for which a computer will output a strictly incorrect - not just imprecise - result because its real solution is uncomputable. This does not happen in the naturals or rationals.
lmm 8y ago
> Suppose we are only working with rational multiples of Pi. How would you start? You can't put a rational multiple of pi exactly into a computer. You can't get a rational multiple of pi exactly out of a computer. You can calculate rational numbers on the computer and scale them by a factor of pi outside the computer, if you really want, but that doesn't mean the computer is working with irrationals. > It's not as though countability lends itself better to programming. Huh? Yes it does; countable sets are still infinite but they're better-behaved than bigger sets.
- mike00632 8y ago
  >You can't put a rational multiple of pi exactly into a computer. Of course we can: "1/2*pi" >You can't get a rational multiple of pi exactly out of a computer. Using the same limitations you seem to have in mind, we can't get one third exactly either. >...countable sets are still infinite but they're better-behaved than bigger sets. Yes, and that is surprising because we do have numerous clever applications that skirt around most of the intractability of exotic real numbers. That is why I think this paper is a bid deal, especially because it points to the cardinality of the real numbers instead of, say, their inclusion of the uncomputable numbers.
  - lmm 8y ago
    > Using the same limitations you seem to have in mind, we can't get one third exactly either. We can work with exact rationals; we can do arithmetic on them and normalize them. In particular, we can compute whether two rationals are equal (in finite time), which we can't do with reals.