- #1
- 10,877
- 422
If G and H are groups, and f:G→H is a group isomorphism, then every theorem that holds for G can be translated to a theorem that holds for H, by changing the domain of discourse for the logical quantifiers [itex]\forall[/itex] and [itex]\exists[/itex] from G to H (or simply changing G to H, if the domain of discourse is taken to be the class of all sets) and replacing each free variable x in the theorem with f(x) (and each constant y with f(y)).
Is the statement above accurate? Would it also be accurate to say that it can't be proved as a theorem in ZFC or even in mathematics, since theorems aren't sets (or categories, or whatever we choose to think of as the foundation of mathematics)? They are strings of text in the formal language of mathematics, so any "theorem" about them would be metamathematics, not mathematics.
Is the statement above accurate? Would it also be accurate to say that it can't be proved as a theorem in ZFC or even in mathematics, since theorems aren't sets (or categories, or whatever we choose to think of as the foundation of mathematics)? They are strings of text in the formal language of mathematics, so any "theorem" about them would be metamathematics, not mathematics.
Last edited: