There are so many problems with both of those statements that it’s hard to know where to start.
How about this: if you represent finite information with reals, your reals will be finite and there’s no problem.
Let’s say my physical system is a rocket launching into space to deploy a satellite into orbit. There are lots of numbers and many equations involved. Exactly none of those numbers we be of the “infinite information” kind. All of the numbers are either measured or derived. Measurements don’t produce an infinitely precise real number.
And of course there’s the whole quantum limit on the amount of information in a given volume of space. So we don’t even need real numbers really.
Even Pi, the example given, doesn’t “contain” an infinite amount of information. It can be represented as the limit of a series using a finite amount of information. In terms of Kolmogorov complexity it contains very little information.
You know, “most” of the reals are uncomputable too. That doesn’t mean the ones we use are all uncomputable.