Is there a single use for this outside of set theory? What changes if I accept/reject the Continuum Hypothesis?

N=1

>>16695593 (OP)
If you allow multisets or labeled numbers, then you can biject P^{1/2, 1/2, 1/2...} with {1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8...} to show that Oresme's proof (the precursor to Cauchy condensation on divergent series) doesn't work in the limit because |N| < |P^N|. For some reason it occasionally makes people seethe when you point this out.

>>16695593 (OP)
Not much. There are no major theorems I know of which are dependent on the CH or its negation. People still study it though because there's always the possibility that down the line it becomes important, like some kind of mathematical analysis method which is contingent on the CH hypothesis being false. In this hypothetical it would become "standard" for the CH to be false, because it adheres to intuition.

>>16695593 (OP)
it’s a pretty basic question when you want to start getting formal about [math]\mathbb R[/math]
>pic
that’s the Generalized Continuum Hypothesis, it’s not so interesting

>>16695593 (OP)
Is there a continuum continuum hypothesis? Suppose the CH is false and there exists a set X such that |N|<|X|<|R|. Then is there a set Y such that |N|<|Y|<|X|?

>>16695593 (OP)
>>16695681
Forgot to add that such a hypothetical is the worst case scenario, and mathematicians are generally unwilling to add any more structure to ZFC. People still study the CH because like I said, there's always the possibility of creating mathematical tools which are dependent on it.

This is where the weirdness comes in: even if such a tool were developed, the CH would most likely still not be resolved. This is because, in theory, there could always exist some modification of the original argument which negates the need to invoke the CH. A full resolution to the CH would only happen in the scenario when there's absolutely 0 possible or known modification to a proof which could remove the dependence on the extra axiom, in which case difficult philosophical questions will start to arise. This has actually happened before with the proof of Fermat's last theorem. Wiles original proof technically depended on the existence of large cardinal axioms, which made it independent of ZFC, it's just you can find a modification of his arguments which do not depend on large cardinal axioms. This does not make his original proof invalid. Here's an explanation:
>https://blog.computationalcomplexity.org/2014/01/fermats-last-theorem-and-large.html

>>16695702
So enlighten me about this basic question, Mr Know-it-all.

>>16695593 (OP)
this looks like a first order recurrence relation lmao.
just take log base 2 and linearize and solve it kek

>>16695593 (OP)
It's a fake question arising out of Georg Cantor's religious schizophrenia - the same goes for all of "set theory" too, of course.

>>16695984
the basic question is “is there anything bigger than [math]\mathbb N[/math] but smaller than [math]\mathbb R[/math]”. it’s a pretty normal next thing to wonder about after you’ve seen the definitions (and before you’ve had the experience to judge whether it matters or not)

>>16696180
pray do tell how you know you ain't schizophrenic as well?

>>16696209
>n-no u!!
Pathetic

>>16695593 (OP)
Yes, General Topology (think Tychonoffs theorem and look up set theoretical topology).

>>16695593 (OP)
>What changes if I accept/reject the Continuum Hypothesis?
Nothing, clearly whatever a set with cardinality between the naturals and the reals is gonna be too specific and pathological to have any uses. We have a fuck ton of results with pathological sets considered already like the Lebesgue integration theory.

>>16696201
This thread is about non-set-theoretic consequences. Work on your reading comprehension.

>>16696312
Tychnoff’s Theorem relies on the Axiom of Choice and has nothing to do with the Continuum Hypothesis.

>>16696529
how about you reading-comprehend the instructions on your AIDS medicine you fucking queer