Are most reals fake? Does it make a difference?

Question

There are uncountably many reals. However, there are only countably many definable numbers. Thus, almost all reals are undefinable. Undefinable means that the shortest representation of that number requires infinite bits of information. This seems very strange because anything representing infinite information fundamentally contradicts physics. So I wonder: Are all undefinable reals "fake" because they have to be irrelevant to everything in our universe?

Furthermore, I wonder if there are any applications in physics of "real" reals? Does it make any difference if you build a theory of (quantum) physics solely with numbers that represent at most finite amounts of information?

Answer 1

Numbers are logic objects, like operators ("add five to the following value, then square it"), operations ("add the next value to the previous value"), and so on. Numbers can work like operators when used as coefficients on variables ("multiply the following value by 5").

Mathematicians care about logic objects.

Physicists care about physical processes. Numbers are logic objects that we can use to manipulate symbolic representations of physical processes or their characteristics. This lets physicists infer the nature of other physical processes when we translate back from our symbolic representational abstracted approximate model of the universe to statements about the real physical universe.

There is no such thing in physical reality as a $5$, just like there is no such thing as a $+$ or a $\int$. Thus it doesn't matter how much energy a $5$ has vs how much energy a $5.01$ or a $5 + 10^{-10^{100}}$ or "the smallest undefinable number larger than 5" has. None of them are physical processes, so none of them have any amount of energy, not even $0$.

Answer 2

This seems very strange because anything representing infinite information fundamentally contradicts physics.

Mathematicians over the past couple of centuries have had a lot of fun exploring the implications of real numbers and real analysis, but none of that existed until physicists invented it. They invented it because they needed it to describe natural phenomena. I don't think you can reasonably argue that something "contradicts physics" when so much of physics is built upon it.

Maybe, some day, somebody will come up with a convincing argument that space and time actually are discrete phenomena at some finer scale than we have probed to date. But, that's not where we are today, and even if it does turn out that way, real analysis most likely will continue to be a valuable tool for understanding things on the same scales that we observe today.

Answer 3

If every number you can write or express, like 0.3432 and π are definable, then yes, you don't need the undefinable numbers. But who knows, they may be defined in the future.