User blog comment:P進大好きbot/List of common mistakes on formal logic appearing in googology/@comment-27513631-20180611232757/@comment-35470197-20180613120520

> The vast majority of mathematicians~

You mean standard natural numbers, right? I could not understand why you think that high-end googology should define such ones. It depends on what rules of googology you are enjoying. (I would not like to restrict googology in my blog post as long as it deals with well-defined natural numbers, because I love ANY natural numbers.)

> Points you in the direction of type theory~

Ah, you meant "we can use other systems than axioms" but not "I can construct mathematics from the foundation without axioms", right? Then it is ok. Sorry for misunderstanding.

> Mind, I'm being a bit of a pedant about the distinction between 'rule' and 'axiom' - an axiom is just a rule with no requirements. (You could do similarly for, like, anything in FOL, by having 'true' be required for a rule.)

In my blog post, I used the terminology "rules" for those on googology, but not those on deduction (i.e. axioms, inferences, and so on). For example, "let's define a computable large number" is a rule which forbids uncomputable large numbers. Do you have any other better word? Regulations?

> Well, yeah. Of course there is.

Ok. I thought that it would be better to avoid using so much mathematical symbols for readers convenience, but it might be a bad way.

> Mind, you could define~

Right. I sometimes do so when I explicitly define a large number. But we should not do so implicitly, because it allows us to define any not-well-defined natural numbers like \(n=10^{100} \wedge n=10^{100}+1\) as \(0\).

> In subsection 8,~

Wow. Thanks. It is seriously bad. I should fix it as soon as possible.

> In subsection 10,~

Hmm... But it is clear that when a statement on SOL VS FOL is true for full semantics, then it is concerning full semantics, isn't it? (There are other semantics than full and Henkin, and hence I would not like to declare them all.) I think that it is enough here, because I just listed reasons why the "strength" is ambiguous.

> given that you're trying to correct common mistakes, clarity seems desirable.

Ok. I will write in that way later. (Sorry for the laziness of the writing way. But the editor tool of this wiki is somehow bad for me... I could not see the result of the conversion of the latex symbols before submitting... Is it a common problem? In googology wiki for anoother language, I could check the latex symbols before submitting.)

By the way, how about section 4? I wonder why you said that there is no maximal incomplete axiom. Is my proof wrong?