>>13794340So metatheory is used by people who want to play games with standards or drag their heels on defining standards.
There is one "standard metatheory" and that is first order logic inference rules.
The axioms are a parameter; unless stated otherwise the "standard library of axioms" is ZFC but in reality nobody believes AC and everybody uses AC to light up ZF like a Christmas tree Satan style \_/ \_/
also "real math" never uses foundation / regularity
this is one of those pedagogical glitches
it's actually an issue in education philosophy
when do you let people play with the rules
when do you force people to play by the standard rules in order to keep them from getting tied up in knots
when you hit model thoery
you're already at Graduate level
so mastery of the current level of standards for much of the math curriculum is assumed tacitly
