Quinn, thanks for posting this. I do not see this as an "argumentative" topic. It would be extremely surprising and disruptive if the proof is valid, but the author has a legitimate-seeming mathematical paper, and the appropriate response is to read it and evaluate its validity. As to why Martin Davis would regard the claim as dubious, I think it's just that over a period of about 100 years, no contradiction has been found and strong intuitions have developed that inaccessible cardinals are consistent and. Equivalent hypotheses have also been used in algebra lately, as you note.

The process of building the new set at successor stages can also be described as taking a set which is Cohen-generic over a countable collection of dense sets $D_{(Y,n)}$, defined by the same criterion used to build $X$ by initial segments. Therefore, assuming MA, the same argument can be used to get $2^{\aleph_0}$ independent sets.

Very good, thanks! If we want to start with an atomless algebra, then I think we can take $B$ to be the boolean completion of Cohen forcing, then fix an infinite maximal antichain in it, and use that in place of ω above. Then build the $\aleph_1$ independent elements using your same procedure, taking sums of the chosen subsets of the antichain.

Honestly I don't see the difference between "notational systems" for natural numbers and something like von Neumann naturals. When you use Arabic numerals, you construct correspondences between the elements of your notational system and some other objects. This is what you do with von Neumann as well when you measure the size of a set. Traditionally the distinction of ordinals and cardinals is one of order (i.e. I am 5th in line) vs. amount (there are 7 people in line), and these are old common sense notions.

Also, "finite ordinal" is certainly not modern. The von Neumann ordinals are, and perhaps the terminology, but "numerals" and "counting numbers" are ancient. Von Neumann ordinals are just one very useful kind of numerals for set theory. Arabic, Roman, Chinese, Mayan, etc. numerals are other equally valid numeral systems for the finite case.

I am not concerned in this question with infinite sets. I want to know if anyone before Dedekind proved that if you can count a set with numerals and reach a stopping point (i.e. count the oranges in a sack), then such a set cannot be put in bijection with a proper subset.