1
$\begingroup$

Suppose $T$ is a first-order theory whose signature contains $(+,\cdot,0,1,<)$ as well as a unary predicate $R(x)$. Suppose every finite subset $S \subseteq T$ has a model in which the set of elements satisfying $R(x)$ forms a substructure isomorphic to the field of real numbers. Does it follow that $T$ itself has a model in which the set of elements satisfying $R(x)$ forms a substructure isomorphic to the field of real numbers?

$\endgroup$

2 Answers 2

5
$\begingroup$

No. Suppose the signature of T contains a distinguished symbol $\omega$, and $T$ contains the statements $R(\omega)$ and the infinitely many statements $1+\cdots+1<\omega$. Then any finite subset of $T$ has a model where $R$ is isomorphic to the reals and $\omega$ is interpreted as some large enough real. But in any model of the entire theory $T$, $\omega$ has to be interpreted as something larger than any real number.

$\endgroup$
5
$\begingroup$

No. Just let the signature contain lots (more than cardinality continuum) of constants and let the axioms of T be that all the constants are different and R holds for all of them.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.