Well great. Then where are you? You are precisely nowhere. You haven’t disproved the results in the system and you haven’t proved anything else.
If the universe indeed uses real values, it's possible that it's more powerful than Turing machines.
However, this doesn't mean that TMs can't operate on such numbers. You can compute the fact that pi + pi = 2 * pi even if you can't enumerate all of pi's decimals.
> However, this doesn't mean that TMs can't operate on such numbers.
Well, it kinda does. For example, you can't compute the sum of two arbitrary real values in a finite number of steps.
That being said, it's possible that the universe actually is not "running" on real numbers. This would also mean that our physics built on top of calculus is just an approximation.
This has always fascinated me. You can't really construct a meaningful amount of calculus without having real-valued variables. The intermediate value theorem depends on it, and it's one of the foundational parts.
I tried to construct something like "bounded error rational calculus", by trying to use rational ranges instead of real values and then constraining the range size. I was able to recover _some_ calculus, but I really have not studied it in enough details.
For example, let's take two arbitrary real numbers, call them x and y. They're almost certainly irrational uncomputable numbers, of course. Now, what is their sum? It's x + y, of course - I can tell you that without doing an infinity of computational work. Now, is this a useful representation? It very much depends on the use you have for it. For two unrelated uncomputable numbers, their sum is almost certainly uncomputable anyway, so there's not going to be any meaningful representation of it either. However, we can still derive useful properties about it, and use it in physical simulations as we need. For example, if both x and y are positive, I can tell you that x+y is greater than either of them, so I can also tell you that the electromagnetic force between two particles that are at distance x+y will be weaker than the force between two particles at distances of just x or just y.
So, what would it mean for the universe to "compute the exact value" of these two, or of their sum? What can physics do, assuming that some aspects are indeed continuous, that a simulation running on a TM actually can't?