Is this what mathematicians actually believe?
https://www.youtube.com/watch?v=SrU9YDoXE88
>>8009205
>vsauce
>mathematicians
no
Honestly? No, it's not.
That video is taking a simple concept and making it way more fucking convoluted and idiot sounding than it needs to be.
We believe in countable infinity and uncountable infinity, and thanks to Cantor's proofs, there can be nothing in between and nothing greater than uncountable infinity.
Countable infinity is the number of elements in the set of all natural numbers. To determine if a set has countably infinite elements, it suffices to see if there exists a bijective correspondence between that set and the set of natural numbers (hence the name 'countable').
Uncountably infinite sets have a bijective correspondence with the set of real numbers. That just means that there isn't a finite number of elements in the set, there IS an order to the elements in the set, and yet there is an infinite number of elements between any two elements in this set. This clearly doesn't meet the definition for a finite set, or a countably infinite set, and yet we can prove such sets exist by means of Dedekind cuts. So we call these sets uncountably infinite sets.
There's slight nuanced definitions like "at most countable", etc., which change the wording a bit, but that's basically it.
That's all there is to it. Very simple concept. Does not need 23 fucking minutes to explain.
>>8009227
>we "believe" in the infinity and power set axioms
>cantor proved the continuum hypothesis
>very simple concept
and you get it so wrong
>>8009227
>uncountable means equipotent with R
nigga what are you doing stop
Wildberger is right
>>8009227
He doesn't make it sound convoluted at all. He does make it sound a lot more interesting than it is, though.
>>8009227
What about a subset of R like (-1,1)
Is that countable or uncountable?
>>8009318
exercise: can you find a bijection between (-1,+1) and R?
>>8009391
Ayy fuck you, I'm just a freshman. How the fuck. I take number theory next year, or is this analysis? In that case then in next next year.
>>8009227
>thinks it's very simple
>completely misses the definition of uncountable
>doesn't understand that that alephs higher than 1 exist
>>8009227
>bijective correspondence is literally how I define every size of every set anyone can imagine xD!
Did you finish your first semester of mathematics, drop out and then decided to start a career of shitposting on le chan? Because that's awfully much what this looks like.
>>8009444
Yeah, that's when I stopped reading.
>>8009743
For raw sets, that's the only common notion of size. For subsets of, e.g., measurable spaces, there are obviously other notions of size.
>>8009234
This.
I hate it when people lecture as if they know what they're talking about when they are actually not only ignorant, but wrong.
>>8009227
If you're trolling, hats up.
If not, let it be known that the video is fairly clear for its standards, P(R) is larger than R, the order (or any order at all) on R is not by any means a requirement for an infinite set and your naive definition of there """exists""" Dedekind cuts is surely not shared by all mathematicians.
>>8009205
probably
>>8009205
Math is so gay
>>8009205
you betcha
