> in short, if it’s convenient, you can make up your own rules.

Yes.

People find it confusing that there is no simple model that encapsulates arithmetic. Fields do not capture it in its entirety. The models of arithmetic that describe it end up being extremely complex.

Arithmetic is ubiquitous in proofs of other things, and people like the author of this blog cannot get over it.

Reality is weird, inconsistent, and weirdly incomplete.

Get used to it!