Genady

I can have a line segment and, using a compas and a ruler, make a 1/3 mark on this segment. Doesn't it measure 1/3?

I don't see what this game has to do with the OP. I have specified my question here:

Yes. r = 1/Pi for example.

It depends on r, of course. But even when it is irrational, any physical measurement or computer representation can only give a rational approximation of it.

Perhaps I need to make my question more specific: Are irrationality of a number or uncountability of a set used in physics to derive a result, to prove a theorem, to formulate a model, etc.?

I don't see any role of uncountability is this experiment. In fact, I don't know of any experiment where an experimental result of measurement is an irrational number.

These are mathematical examples. I'm looking for an example in a theory in physics. Pi, e, sqrt of 2 appear in physics many times. Of course, they are irrational. But would anything in physics change if they happened to be rational? Does the fact that they are irrational play any role in physics?

I am looking for a difference, in physics, between infinite countable and infinite uncountable.

I don't think so, because in this case the number of steps in any finite time would be finite.

Does the fact that set of real numbers is uncountable, as opposed to rational numbers, for example, play any role anywhere in physics? More generally, do infinite cardinal numbers play any role anywhere in physics? For example, in an infinite dimensional Hilbert space, does it make a difference if the number of dimensions is countable or uncountable?

Since everything that has dimensions and/or duration can be, in principle, a ruler / a clock, it equally affects everything like this.

You mean, non-Minkowskian rather than non-Euclidean. The latter could apply only to space. Alternatively, it would be fair to say that space-time is Minkowskian but gravity affects all rulers and clocks in such a way that measured distances and times are distorted. Einstein field equation describes these distortions.

Sounds like dogma to me. Let's call it, research program. You seem to like this research program. I seem to dislike it. A matter of taste. Which research program will go farther, is a different matter. Yes, you don't. But Lorentz Jr. does.

No, it would not. Perhaps you misunderstood what 'the "proper time" is the same for any observer' means. What do you think it means? In my understanding of it, time dilation is 'real'. Yes, your understanding of proper time.

Here is a little thought experiment. One of the twins stays on Earth. Another goes away for space travel, visits many places, experiences many environments, etc. When the traveler finally comes back to Earth, after 30 years, they discover that somehow, miraculously, they aged exactly the same! Wouldn't it be strange? Wouldn't it require a mechanism to explain how such synchronicity could happen?

See QFT and Standard Model extensions. See QM. See various inflation models. See SR. Anyone is allowed to come up with a hypothesis. Everyone is allowed to ignore it.

The current hypotheses answer these questions in various ways. As there are no data, no one is more correct than others. If you want to know their answers, you will need to study.

Only if it has support by relating to what we do know about matter, radiation, space and time.

There are several hypotheses regarding what led to the Big Bang, but there are no data yet to test these hypotheses.

Matter and radiation. Mostly radiation. But it was not a "point".

I am sure this Burundi Presidential Palace has cameras all over:

Which island is it? (Maybe I've been there.)

I think the answer is here: The 15 Poorest Countries in the World - WorldAtlas

Free PDF of this book downloadable here: Download PDF - Einstein In Matrix Form: Exact Derivation Of The Theory Of Special And General Relativity Without Tensors [PDF] [561fu48ukqq0] (vdoc.pub)

The main point, to me, is that mathematics has explanatory power. Regarding math vs physics, they are perceived differently on the level where our perceptions are connected to our senses. As we go down the turtle levels, the distinction becomes less evident. I see a possibility that turtles are not all the way down, but rather the last level of turtles is just math. There, the mathematical explanations are the explanations.