It's good to finally have something up we can use. You have some very interesting questions there about the way we "describe" dimensions. I use the word describe because I believe that we do not build dimensions per say, but rather, we build descriptions of the dimensions. The problem I believe is that we mix description with described object. Dimensions allow us to be able to have something we can imagine about space [not the 3 D space]:)
The real question which you raise is, can we have a different description that works as well or maybe better? Another way of asking the same question is: what is wrong with the current description? Does it for example allow us to clearly define infinities? I think not and this ties back to our last conversation about the set of "all" infinities. What is it? how is it constructed? Does such a set even exist?
PS: Adding to your list of some of the things we establish last time, we "proved" that there are as many rational numbers as there are irrational numbers by creating injections both ways.
We did not quite reach a conclusion on the number of infinties as we tried to define what the set of infinities could look like. One of the ideas was that it is itself an infinite set...puzzling...