Date: Feb 3, 2013 8:31 PM
Author: Graham Cooper
Subject: Re: If 2 systems represent the same sets do they prove the same theorems?

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.

Herc
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