I’m now working on the inf set section of the course. There are new things that were not included in the Infinity article so I thought it was better to write them briefly here before publishing the course sometime in the future. I think, based on the messages I get, it will be more beneficial for me to focus more on math/phys topics.
It was shown in the article that the process of counting never diverges. In set theory the infinity axiom is included for this purpose, it ensures there is a set containing all natural numbers. The cardinal number for such a set is not derived from proofs. As discussed in the article if real and natural numbers diverge similarly, then the diagonal method doesn’t produce a new real number. However, divergence in this proof is ambiguous it is not clear whether this is a continuous process or not. Or whether there is a neighbourhood around inf, and if there is, the process of entering that area is unclear. the proof does not provide insight into the process of divergence. It assumes that as given without outlining its properties. It more or less assumes the cardinal number of the set of N is similar to finite sets in the sense that it gets bigger and bigger and eventually, it diverges. the ambiguity of the divergent process is the main issue with the proof. When it is unclear how natural numbers diverge it will also be unclear how the 1-1 correspondence between different inf sets should be drawn. This proof is a big step toward understanding infinity but the infinity article argues that this approach is incomplete.
According to the article, the set of N should be accompanied by a mechanism that pushes finite to the inf realm, something similar to the mechanism that is included in the projective geometry (so the infinity axiom should be expressed in a different way). So the cardinal number should be based on that mechanism. for example, in proj geometry, the cardinality (of the proj line; for example R) can be based on angles and not numbers. So set theory can be expanded beyond ZFC framework where the cardinality of inf sets is a function of the gate mechanism. The issue here is that the gate mechanism does not fit into the axiomatic framework and is more suitable for OOP structures.