[FOM] Question about Conservative Extensions

[william tait]

Sent from my iPhone

On Sep 13, 2012, at 11:10 AM, Richard Heck <rgheck at brown.edu> wrote:

>  If T2 is a conservative extension of T1, can every model of T1 always be expanded to a model of T2?

If 'expand' means they have the same domain, which I usually take it to mean, the answer is no: let T1 be complete number theory and T2 the result of adding all the sentences n< c for each numeral n, where c is a new constant. If you will allow the domain of the 'expansion' of the model M of T1 to be an extension of the domain of M, the answer is 'yes' by a simple compactness argument: 

If T2 does not have a model that is the expansion of a supermodel of T1, then it is inconsistent with a sentence phi(m) which is true in T1, where m is a list of names of elements of M not in the language of T2 and phi is in the language of T1. So T2 implies forall x not-phi(x), a sentence in the language of T1, which is false in M. So T2 is not conservative over T1.

Bill