Funnily enough, given your username, if you let de Sitter space evolve, it will -- literally -- grow to infinite volume.
Barring some source of gravitation greater than that implied by studies of the energy content of the observable universe, and in particular under the \Lambda-CDM model, our patch of the universe will ultimately evolve to something so similar to de Sitter that it should (again, literally) grow to infinite volume.
I don't think many physical cosmologists would complain about determinism; this is in part reflected in initial values surface formalisms.
Your last two points have been the subject of substantial investigation; you might enjoy some of the pointers to primary sources in this article: https://www.wikiwand.com/en/Digital_physics
As noted therein, it is hard to divorce real numbers from working models that fundamentally explain (and predict) experimental and observational results at the energy scales we have access to.
("Fundamentally explain" as in "fundamental theory" as a term of art, as it were; contrasted with things like effective theory or classical theory or reduced theory.)
Although I would not word it like your last sentence, you hit the nail on the head by saying that we have good evidence that there are some _local_ properties that do not change with a displacement in space or in time. That's the principle of relativity at work. Unfortunately when you accept that one of those invariants is the local measurement of the speed of light (or indeed the speed of any object with zero rest mass), you run smack into infinitesimals that under detailed experiments to date actually look physical.
Well, this de Sitter space would "--literally-- grow to infinite volume" only in theory. If you would actually try to build such a thing I'd say the probability that at some point you would remain without building blocks is very very high..
Mathematics is a great tool, but it has the drawback of not being equipped to distinguish reality from fiction..
The way I've reached my oppinions is by asking myself a simple question ( although the answer is obviously not that simple ) : How would I build a Universe ?
You know the saying, "If I cannot create it, I do not understand".
From this perspective infinity is not an option and the "spacetime" doesn't really make any sense.
Time would, at most, be a way to measure the transition period from one state to another, but it cannot influence how your system works. The output should be the same regardless of how much "time" you would need for one transition...
dS space is really simple; you only need a cosmological constant to power a uniform metric expansion, and no matter to retard it. The far future dS space approximation has only localized matter, namely mostly isolated gas molecules and (swarms of) black holes, each occupying their own Hubble volume, as all their non-gravitationally-bound neighbours have disappeared across the horizon.
Unless something really unexpected arises to retard the metric expansion of space -- and many working scientists are looking for that, since it would be amazingly cool -- that we see right back to the surface of last scattering, the "in theory" part looks very sound.
"If I cannot create it, I do not understand" -- well, get yourself a very fine spectroscope and a monochromatic radiator (or an ideal blackbody, or anything with a highly predictable emissions and/or absorption spectrum) and probe the infinitesimals in the Poincaré group, which is the isometry group of Special Relativity (and which, incidentally, is what fundamentally defines Minkowski spacetime). https://www.wikiwand.com/en/Modern_searches_for_Lorentz_viol...
An experiment or observation inconsistent with infinitesimals, or preventing the far future dS state implied by observed energy-densities, would be groundbreaking.
Although your objection to a physical infinity, or physical infinitesimals as their complement, are aesthetic ones that conflict with experiment to date, don't let that stop you from thinking about, or even proposing, novel, objective, and repeatable experiments that probe the abolition of real numbers.
I'm sorry, but you're using all these terms that I don't really understand and I don't have time nor interest in studying.
This doesn't mean that I can't think about things. I may not get far, but that's not the point. Actually I find great pleasure in reflecting about the more subtle things of my existence, as much as I can perceive them, and the pleasure is greater as I'm less biased by the knowledge of others. It's just me...
I do believe however that the truth can be rediscovered and there are more paths to it..
Thank you anyhow for your information packed replies. Don't really know if they were meant for me or for you but doesn't really matter. I probably remained with something.
Barring some source of gravitation greater than that implied by studies of the energy content of the observable universe, and in particular under the \Lambda-CDM model, our patch of the universe will ultimately evolve to something so similar to de Sitter that it should (again, literally) grow to infinite volume.
I don't think many physical cosmologists would complain about determinism; this is in part reflected in initial values surface formalisms.
Your last two points have been the subject of substantial investigation; you might enjoy some of the pointers to primary sources in this article: https://www.wikiwand.com/en/Digital_physics
As noted therein, it is hard to divorce real numbers from working models that fundamentally explain (and predict) experimental and observational results at the energy scales we have access to.
("Fundamentally explain" as in "fundamental theory" as a term of art, as it were; contrasted with things like effective theory or classical theory or reduced theory.)
Although I would not word it like your last sentence, you hit the nail on the head by saying that we have good evidence that there are some _local_ properties that do not change with a displacement in space or in time. That's the principle of relativity at work. Unfortunately when you accept that one of those invariants is the local measurement of the speed of light (or indeed the speed of any object with zero rest mass), you run smack into infinitesimals that under detailed experiments to date actually look physical.