[FOM] Question about entailment between sentences belonging to different theories

[Richard Heck]

On Mon, 2003-06-09 at 16:06, addamo at wp.pl wrote:
> Let theory T1 be the arithmetic of natural numbers with individual
> variables: [description of language and axioms followed]
> Let theory T2 be the second order arithmetic of real numbers with primitive
> concepts concernig individuals: +, *, < (less then), =, 0, 1; and variables
> for individuals x,y,z, ..., and for sets X, Y, Z, ..., functions
> F,G,H,.....; symbol # (x # X reads elemnt x belongs to the set X); symbol Q
> for null set [Axioms followed].
> I hope this rough description will suffice to put more precise the question:
> in which sense could we say 'R16 [of T2] entails N7(phi) [of T1]' or simply 'R16 entails
> N7".

Unless I am misunderstanding the description of the theories in
question, then there does not seem much of an issue about how to
understand this kind of question, because the language of first theory
is contained in the language of the second. (If the language of the
first theory were not contained in that of the second, then things would
become a good deal more complicated.) If so, then one can simply ask
whether N7 is provable in T2. Of course, since N7 here is induction and
so is a schema, one really has to ask whether all instances of N7 are
provable in T2. I'll ignore that fact henceforth. I'll also continue to
assume that the question here is really about provability, not
entailment. Nothing I'll say will turn on this matter, however.

Now, what I just said might not answer the original question, since what
was asked was how to understand the question whether, say, R16 entails
N7. One might try asking the question this way: Can one prove, in T2,
that R16 --> N7? But since R16 is an axiom of T2, that question is just
equivalent to the question whether N7 is provable in T2. One might then
try asking whether R16 --> N7 is provable from the axioms of T2 besides
R16. But that's again an equivalent question. The more general point is
that the question whether N7 is derivable from R16 has to be asked
against some background theory, and whatever the background theory might
be, the question can simply be put in the form: Are all instances of R16
--> N7 provable in that theory? (Or, if the question is about
entailment, are all of its instances true in all its models?) 

If that doesn't answer the question, please clarify.

Richard Heck