Having reached the concluding portion of my first course in real analysis, one subject that I feel was not adequately addressed was the issue of cardinalities.
This is a subject I was interested in before taking this course, particularly in some of the stranger aspects and implications of set theory and incomputable numbers. While I personally philosophically question the validity of claiming the existence of something you cannot write down, I understand formally the arguments that leading to the conclusion that there are more real numbers than there are computer programs; i.e. I can translate each possible finite length string of a finite selection of characters into an integer, and then show there is no bijection from the integers to the reals).
My question here is if we are as limited in what we can write down as is typically assumed, namely: this proof assumes there are a finite number of characters. But is this in fact true? Given a piece of paper and a pen, are there really only a finite number of ways to arrange ink molecules on the paper to create a characters? (From my understanding of quantum mechanics, molecules are not limited in possible arrangements in position space; rather it limits the accuracy with which you can know a particle's state in position-momentum space, though this is besides the point).
If there are, in fact, an uncountable number of characters we can write on a paper, does this make every real number computable? Can we make compilers that could handle this, or does this violate some part of the definition of a Turing Machine (with which I am admittedly only vaguely familiar).
I tried a quick google search for this idea, and nothing seemed to come up, so I am curious if this has been explored. I would at the least be very curious to learn about theories of computation under the assumption that we can write programs using an uncountable, or greater, number of characters, and how that might actually work.
Thank you for your answers.
