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> The work of Cantor and beyond regarding infinite sets is generally taught as mathematical "fact", as fundamental as 2+2=4, but unlike 2+2=4 is there any pract

> The work of Cantor and beyond regarding infinite sets is generally taught as mathematical "fact", as fundamental as 2+2=4, but unlike 2+2=4 is there any practical constraints or applications around it? Yes. There exist libraries that allow us to do arithmetic with arbitrarily large integers with exact precision in a computer, but none that allow us to do arithmetic with arbitrary real numbers. This is because you cannot ever find an encoding that assigns each element of an uncountable set a finite-length string, but you always can find one such for countable sets, no exceptions. So in practical day-to-day, the programmer (and computational modeller) has to settle for all the quirks of imprecision when dealing with floats, and this imprecision makes a lot of algorithms more complex and limited than the mathematical computation they are trying to model.

WastingMyTime89 5y ago
> that allow us to do arithmetic with arbitrary real numbers Of course there are. You can symbolically encode and manipulate real numbers with exact precision. Sure it might become unwieldy and is fundamentaly constrained by the physical limits of your computer but the same is true with large integers computation. People don't do it usually because it's useless not because it's impossible. Bounded precision is fine most of the time.
- anchpop 5y ago
  That doesn’t let you work with arbitrary real numbers because the vast majority of real numbers cannot by symbolicly encoded, even on a turing machine with infinite memory
  - Firadeoclus 5y ago
    If they cannot be symbolically encoded why would you ever _need_ to encode them?
    - anchpop 5y ago
      Not being able to do something has never stopped people from needing to do it :P That said, with Google's paper "towards an api for the real numbers", you get close enough for many practical purposes

WastingMyTime89 5y ago

> that allow us to do arithmetic with arbitrary real numbers Of course there are. You can symbolically encode and manipulate real numbers with exact precision. Sure it might become unwieldy and is fundamentaly constrained by the physical limits of your computer but the same is true with large integers computation. People don't do it usually because it's useless not because it's impossible. Bounded precision is fine most of the time.

anchpop 5y ago
That doesn’t let you work with arbitrary real numbers because the vast majority of real numbers cannot by symbolicly encoded, even on a turing machine with infinite memory
- Firadeoclus 5y ago
  If they cannot be symbolically encoded why would you ever _need_ to encode them?
  - anchpop 5y ago
    Not being able to do something has never stopped people from needing to do it :P That said, with Google's paper "towards an api for the real numbers", you get close enough for many practical purposes