I haven't heard of the first two, but an astronomy teacher I had in high school once explained continuum hypothesis to my class. He was just doing it for fun though, so we weren't tested on it or anything. All I can say about it is that Georg Cantor had a little waaaaaaay too much time on his hands.

What Georg Cantor did was truly oundational work. Modern math would not be possible without his contribution. Topology as we know would most definitely be much more anemic without Cantor's insight into the nature and notion of infinity (or rather infinities).

So I don't see it as a waste of time!

I don't think we currently have a way to reduce such things down to math and logic as of yet; there's too many variables. We may one day, but we do not currently.

way over my head but my laymen response to:

I see math as a tool which approximates reality, used at the scale we describe things to (without renormalization) , in that case I feel the existence of an empty function to be similar to trying to demonstrate nothing using examples of something. its null I am a person who does not believe numbers exist as actual things... or rather when they do exist they take on different forms eg the wiring of the brain for the concept of the number 1, or subjectively dividing things made of millions of atoms into groups and numbering them where their order is arbitrary.

axiom of choice i think you would have to prove that things have identity which seems to either be objective in terms of math but subjective in terms of the mind to manipulate once again the tool which we created.

continuum hypothesis well I don't understand this very well but all I can say is it appears to be somewhat a similar thing to describing fractional dimensions like fractal geometry, which seems to exist in nature and can be described but is once again another invention of our tools to describe it... You have to look closer to see something, the closer you look the more you add to what you are observing until all you are looking at is more of yourself (or the measuring device, or the function)

I am not good at math so im sure everything I say is probably flawed or me misinterpreting these "problems"

what a load of bologna, silly seal :P

I see these axioms in the same way I see the postulate of parallel line (thus producing classical Euclidean and non-Euclidean geometries)...just axioms. One can include or reject them as one sees fit. But the deeper point is that there are more "deep" ideas out there. It is a bit like peering into the far, far corners of the Universe, and stranger stuff begin to appear!