This is pretty far into crank territory. One of the issues with this paper is simply a misunderstanding of terms. The word "real" as in "real numbers" is jargo

This is pretty far into crank territory. One of the issues with this paper is simply a misunderstanding of terms. The word "real" as in "real numbers" is jargon. It's not about whether or not you believe in them or even whether they are representative of reality. You can think of them entirely in logical terms (whether logically constructed with Cauchy sequences or Dedekind cuts) and then the disagreement is about the axioms of set theory. Numbers are just a theoretical framework, that is how they are used and "believed in". If you want to say that real numbers are absurd or not fitting of our world then simply propose an alternative theory that we could elect to use instead of real numbers. Nobody is adamant about the idea of all real numbers being "real" in every sense of the word. Furthermore, a theory of numbers with a restriction on the size of the set is arbitrary. Cantor's diagonalization argument and construction methods still exist. To exclude uncountably infinite sizes of sets would make as much logical sense as restricting the total number of numbers to something finite, e.g. "there can be no numbers greater than a hundred billion, try not to think of a hundred billion and one because that is bullocks!". An interesting example of irksome numbers being remedied with better theories is the concept of infinitesimals in calculus/analysis. The theory of Nonstandard Analysis provides a rigorous definition of infinitesimals in terms of more familiar numbers. https://en.wikipedia.org/wiki/Nonstandard_analysis https://en.wikipedia.org/wiki/Nonstandard_analysis