Genady

This is what we are talking about for the last few posts, isn't it? Yes, different and completely imaginary story. Not interested.

I disagree with this translation. It assumes that there are 'real' space, time, or other variables. IMO, all these variables are components of our models. They are what they need to be for our models to work. That's why my question rather is, is it important / used / implied anywhere in our current models if the variables have or don't have the power of continuum. I am sorry, but it doesn't say much to me. It does not matter for being able to represent all rational numbers. Any finite size will do. Yes, this is the necessary condition for being able to represent all rational numbers.

Why? I don't see that my previous statement depends on the grid resolution. I see only two conditions for the grid: a) the steps have a finite size, and b) there are infinitely many of them.

I take it back. In an infinite discrete grid of finite cell size, all rational numbers can be represented.

In this case places represented by rational numbers would not apply too.

Which units are fundamental, and which are derived, is a matter of convenience. There is nothing fundamental in this choice.

I think I understand this intuition. To rephrase, if variables in equations were rational instead of real, some terms would be undefined. This is a good answer to the OP question. Thanks. +1

In general, no, this is not the only thing that matters to me. In this thread, yes, it is. Your previous posts are interesting on their own. We can discuss them in your thread.

@TheVat, I think we can measure 1/3 (or any other rational number) in cases when we can measure by counting. I mean, if we have counted 1000 of something and we know that there are 3000 of them in total, we have measured 1/3, right?

Thank you. Yes, I agree. The numbers we get experimentally are at best rational, but we don't get all rational numbers. But in theory? (Which is the question in OP.) Is it important anywhere in physics theory if a number is rational or irrational, or a set is countable or uncountable?

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?