/sci/ - Science & Math

>>11666003
All sets that contain numbers beyond [math]\mathbb{R}[/math] are just extensions of [math]\mathbb{R}[/math] meant to serve some esoteric purpose, that [math]\mathbb{R}[/math] is not fit for.

[math]\mathbb{R}[/math] already contains numbers incredibly esoteric, like uncomputable and udefinable (in ZFC) numbers.

>>11236883
>Disregarding anything that can't be described by a turing machine in finite time is a very limited view of mathematics.
You've literally never encountered a number like that.

>>11236917
The situation is worse than that. The set of numbers with a finite description is countable. So with probability 100% if you pick a real number at random, it will be an undefinable number.

They can't even be described in finite time/space. I'm not going to be all Wildberger-y and deny ALL non-rationals but if you're trying to say a number exists that doesn't even have a construction you're pulling shit out of your ass. Proofs are constructions get out of here with your santa clause BS.

>>10256717
You don't.

>>10214709
Ha, this is neat. For people who are interested, the problem with this approach is that with 100% probability any real number you choose can't be given a finite description and if you don't have a finite description then you don't have a finite program and the only programs that can be encoded as integers are finite programs. In other words, only a countable proper subset of the reals (that contain the algebraic numbers as a proper subset) are actually definable. The remaining undefinable reals are just that, undefinable. That means that any real number you can describe is definable and the undefinables are pretty much unthinkable numbers and only "exist" as an abstraction that you can't produce any examples of.
While I personally believe the justification for believing the reals "exist' is tenuous I don't personally have anything against them as an abstraction in the same way I don't have anything against using the hyperreals instead of the reals whenever it makes things more convenient.

>>9929483
>Ironic.
Ironic.