3 ms·

I don't think you can define the reals as a mapping from natural numbers, reals are famously not countable.

Tainnor 2y ago
You can't map the natural numbers to the set of real numbers, but you can map the natural numbers to the fractional part of a single real number.
- defrost 2y ago
  How would this work when mapping them to the fractional part of the real number 2 ? Collapsing a countably infinite set to 0 doesn't seem useful or reversable.
  - gizmo686 2y ago
    The integer part is easy, since we already have the integers. Once you have D=[0,1), then you can define R=ZxD. That is to say, this definition of R seperates out the integral and fractional components of every real number.
  - Tainnor 2y ago
    fun n => 0 I'm not sure what you're trying to disagree with here.
robinhouston 2y ago
Fortunately functions from the natural numbers are not countable either. This is Cantor's theorem.
danwills 2y ago
I'm not a set-theorist, but a set-theorist friend of mine once taught me that you can turn a countably-infinite set (such as the integers) into an uncountably-infinite one (like the reals) by applying the 'power set' operation (the set of all subsets). Not heaps sure what this really means with respect to whether 1 the integer is really completely related (as in, equal, or the-exact same-thing) to 1.0 the real though. Kinda seems like it might still need a bit more information to fully identify a real, even when it happens to be infinitely-close to an integer?
- Tainnor 2y ago
  More generally, Cantor's diagonal argument shows that there is no surjective map of a set onto its powerset.
housecarpenter 2y ago
I think you misunderstood the parent comment. They're not talking about defining the reals by setting up a mapping where each real number corresponds to a unique natural number. They're talking about defining the reals by setting up a mapping where each real number corresponds to a unique mapping from the natural numbers to {0, 1} (i.e. a unique binary sequence). The set of all binary sequences is isomorphic to the power set of the natural numbers, which is uncountable.

You can't map the natural numbers to the set of real numbers, but you can map the natural numbers to the fractional part of a single real number.

defrost 2y ago
How would this work when mapping them to the fractional part of the real number 2 ? Collapsing a countably infinite set to 0 doesn't seem useful or reversable.
- gizmo686 2y ago
  The integer part is easy, since we already have the integers. Once you have D=[0,1), then you can define R=ZxD. That is to say, this definition of R seperates out the integral and fractional components of every real number.
- Tainnor 2y ago
  fun n => 0 I'm not sure what you're trying to disagree with here.

robinhouston 2y ago

Fortunately functions from the natural numbers are not countable either. This is Cantor's theorem.

danwills 2y ago

I'm not a set-theorist, but a set-theorist friend of mine once taught me that you can turn a countably-infinite set (such as the integers) into an uncountably-infinite one (like the reals) by applying the 'power set' operation (the set of all subsets). Not heaps sure what this really means with respect to whether 1 the integer is really completely related (as in, equal, or the-exact same-thing) to 1.0 the real though. Kinda seems like it might still need a bit more information to fully identify a real, even when it happens to be infinitely-close to an integer?

Tainnor 2y ago
More generally, Cantor's diagonal argument shows that there is no surjective map of a set onto its powerset.

housecarpenter 2y ago

I think you misunderstood the parent comment. They're not talking about defining the reals by setting up a mapping where each real number corresponds to a unique natural number. They're talking about defining the reals by setting up a mapping where each real number corresponds to a unique mapping from the natural numbers to {0, 1} (i.e. a unique binary sequence). The set of all binary sequences is isomorphic to the power set of the natural numbers, which is uncountable.