[FOM] Question about entailment between sentences belongingto different theories

[addamo@wp.pl]

Dear Professor Heck,

this was my second posting on the theme. In the first one I asked for the
more difficult and general question you have metioned in your mail.
Is it really so that you can get each instance of induction scheme in T2?.
What would it be, for example, with formula phi = x>0 or x=0? Don't we need
to relativize the quantifiers to the other domain (exactly to the set of
natural numbers which is definiable in T2)? If so, the relativized formula,
call it
N(rel.phi) wouldn't be the same formula as N7(phi).

An idea of general setting.
Let's take two theories T and S. Two formulas A of T and B of S. We say that
A entails B iff there exist theory F and transformations tau and sigma and:
tau(A) entails sigma(B) on the ground of F in the normal sense of 'entails'
(where tau(A) and sigma(B) belong to F and are some kind of translations.
For me it could be a generalization of 'normal' entailment relation. If the
theory T were subtheory of S, the tau and sigme would be simply identity.

Adam Olszewski

PS. I am a philosopher not a mathematician.