On Feb 3, 8:31 pm, Graham Cooper <grahamcoop...@gmail.com> wrote: > On Feb 4, 9:21 am, Charlie-Boo <shymath...@gmail.com> wrote: > > > > > > > > > > > On Feb 3, 5:29 pm, Graham Cooper <grahamcoop...@gmail.com> wrote: > > > > On Feb 2, 3:38 am, Charlie-Boo <shymath...@gmail.com> wrote: > > > > > That is, if for every wff w in system A there is a wff v in system B > > > > such that |-w(x) iff |-v(x) for all x, and likewise for vice-versa B > > > > is in A, then do systems A and B prove the same theorems? > > > > > C-B > > > > by extension system A = system B > > > > However, one system may have restricted comprehension on WFF while the > > > other does not. > > > > Herc > > > Are you saying yes to my original question? > > Your subject line I'm ignoring. > > By extension > > (ALL(thm) thm e theoryA <-> thm e theoryB) <-> (theoryA = theoryB) > > ALL(thm) A|-thm <-> B|-thm > <-> > A=B > > However the set of WFF in each may be different. >
> The parameter X limits what you are trying to say > as theorems have higher arity than 1.
X is a tuple where the number of components ranges from zero to infinity with arbitrary cardinality.