[techno_modus]

It would be interesting to learn about some new (unified) theory that can formalize the notion of infinity. Apparently, this can hardly be done on the basis of the classical (Cartesian) view of space which is essentially a box with points (elements).

[Mugwort]

Then John H. Conway's Surreal Numbers might be of interest to you. Donald Knuth wrote a book about the subject and it's actually in the form of a story with dialogue, not your typical math book. I'll attempt a summary. It is possible to build all number systems at once including all integers, rationals, algebraic, irrational, infinitesimal, hyper-reals and transfinite all at once using a single procedure which resembles Dedekind cuts. From that foundation one can recreate many of the results of non-standard analysis. Conway discovered his Surreal Numbers while investigating Combinatorial Game Theory. He wasn't even trying to create a new foundation for mathematics, he was just playing at his games which many of his colleagues looked down upon as useless. Apparently playing games and having fun can occasionally result in extraordinary discoveries.

Conway's left and right "cuts" look like sets but then again I'm not really sure what they are. Is this alien mathematics accidentally discovered on Earth by perhaps the most intelligent Earthling who ever lived. I don't know. All I can say is I took a detour from my usual math studies to look into Surreal numbers and it was an interesting trip. I'm quite conservative when it comes to math. I don't go around preaching Surreal numbers are the way to do math but it is a curiosity none-the-less worth at least the small price of reading Knuth's short book about it from cover to cover.

[voxl]

If you listen to Conway talk about the Surreals you'll note that he considered it a game (in the game theory sense). He was interested in games of a certain kind and what properties they had, a long the way he stumbled upon this game that we now call the surreal numbers.

[pfortuny]

I do not understand your question: a set is infinite if and only if there is a bijection between it and one of its strict subsets.

As formal as it gets.

Edit: (not intending to be harsh, just that I probably lack some context to properly understand what you are asking).

[ngcc_hk]

Is it point or is it continuous line with infinity as an “end”? And there is infinity elements without any point. Or any section of any one of these baselines (and you can have infinite dimension hence many baselines not just x y z, or r/degree etc) can be remap to itself and hence is infinite as well. The point is not the lines, as mapping proved that the points (said the natural no point) is a smaller subset of the real number or the line.

Just not sure about your point.

[sidusknight]

This is why math shouldnt be posted on HN.

[unixbeard1337]

There is worse misunderstanding/bikeshedding upwards in the thread, although it's more artfully worded. And at least one of the replies to this refers to useful resources. This is, in fact, exactly why math should be posted on HN.