2
$\begingroup$

Good day to everyone,

I would like to ask this question. I was wondering what is considered to be the hardest area of mathematics, so I searched in forums and outside of answers like Langlands programe and Cohomology there was this answew called Irational number theory, where it was stated that we basically have no idea of how to even solve the basic questions. Which leads me to my question.

Do we have number systems of higher cardinality then Reals? I don´t mean number systems like Complex numbers, Quartions, ... as all of them have the same cardinality as Reals, I mean number systems like the power set of Reals and power set of that power set etc. And do we do some math(outside of set theoretic math), meaning have big open problems like the millenium prize problems in those number systems? Or is it more or less corect to say that almost all of our current math research is done in Computable numbers and mathematics in those "higher infinities" is too hard for our finite brains?

Thank you for your answer.

$\endgroup$

marked as duplicate by Asaf Karagila cardinals Jun 28 at 18:40

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ The Millennium Problems are all quite concrete problems, and quite within the world of the real numbers (at least cardinality-wise). $\endgroup$ – Asaf Karagila Jun 28 at 18:39
  • $\begingroup$ @AsafKaragila But still may I ask you would you say that almost all of our current math research is done in Computable numbers and mathematics in those "higher infinities" is too hard for our finite brains? I mean the vast majority of problems are undecidable by Turing machine so it therefore must follow that problems with which out current math is occupied with like the Millenium prize problems which lets assume are decidable are much simpler than undecidable problems. Is my reasoning here correct sir? Thank you. $\endgroup$ – Pan Mrož Jun 29 at 10:00
  • $\begingroup$ It's not that, it's just the fact that once you move out of countably generated objects (in the context of topology, separable is countably generated), you find set theory and independence get involved a lot more than what people are comfortable with. $\endgroup$ – Asaf Karagila Jun 29 at 10:11