Everyone wants to solve an open problem. OK every nerd wants to solve an open problem. But whether you’re just some random nerd or the next Godel, this is the web page for you. I have presented several open problems on this site, so post your solutions.
Or partial solutions.
Or just random ideas that might help.
Within the presentation for each open problem, I have given you the ability to post your solution, alongside other solutions. This way many different potential solutions can coexist within the same problem presentation.
This link is an axiomatic system that I have developed to try to address the generalized continuum hypothesis.
It is here that I will actually address it though. https://mathematicsthink.com/welcome/new-axiomatic-systems/#comment-21
Lets examine the numbers of the form $T=b_1,N_2,b_3,b_4,…$, where $T$ has partial depth because $b_n$ is either 0 or 1 and $N_2$ has depth 0. Let us denote the cardinality of the numbers that take this form $\beth_T$.
Theorem: $\beth_1<\beth_T<\beth_2$ implying that the generalized continuum hypothesis is false.
Let’s start listing the values of $T$. We start by changing only the values of $N_2$ and keep $b_1=b_3=b_4=0=…$ The cardinality of this listing is $\beth_1$. We can do this with every potential value for every $b_n$. In some sense we must have $\beth_1$ of $\beth_1$. Does this mean that we would not be able to create a bijection from the set of all $A$ to the set of all $T$, where $A=b_1,b_2,b_3,b_n,…$?
At this point I still don’t know. What’s the difference between that and say, the cardinality of $R^2$, which we know is $\beth_1$.