User blog comment:P進大好きbot/List of common mistakes on formal logic appearing in googology/@comment-27513631-20180611232757/@comment-35470197-20180612013456

> notes that you can work internal to a model and just call it the universe (as long as you're explicit about the language permitted)

Right. As I wrote "You are working on M itself and your definition names a natural number relative to M". (Is it the same as what you meant?)

> notes that in normal mathematics the axioms (for, say ZFC) are all implicit and that in our context we usually mean "additional axioms"

Right. I know the tradition of the normal mathematics, because I am a pure mathematician (and also you?). But some Rayo-like ones are not valid even if we assume ZFC.

> notes that axioms are just a special case of rules (the ones with no requirements) so technically it's possible (easy) to do mathematics without axioms

How could you do so? Of course, you can add deduction rules in order to remove axioms, but it is just an interpretation of axioms.

> notes that a definition is valid if \(\exists !x \phi(x)\) is true, even if not provably so, if you also consider the case where it is false separately

It is just a choice of terminology. I referred to the definition of a definition from the standard textbook by Kunen, who is one of the greatest mathematician. What is your reference?

I would like to call the definability in your sense the namability. Of course it is ok to use the namability. I will add the description. Thank you for pointing it out.

> notes that for pretty much all high-end recursive and non-recursive googology to always define actual naturals, as opposed to model-relative ones, it's usual to restrict attention to models with the actual naturals

What "actual" means here? I could not understand what you want to point out.

> recommends you explicitly state languages separate from their theories and don't restrict attention to set theories only when speaking in generalities

Is there any ambiguity? Why do you recommend so.