3
$\begingroup$

Let $\sf MKCC$ stand for Morse-Kelley set theory with Class Choice. And let this theory be precisely $\sf MK$ with a binary primitive symbol $\prec$ added to its language, and the following axioms ensuring that $\prec$ is a well-ordering on classes.

$\textbf {Transitive: }X \prec Y \land Y \prec Z \to X \prec Z \\ \textbf{Co-Connected: } X \not \approx Y \iff [X \prec Y \lor Y \prec X] \\ \textbf {Well-Founded: } \phi(X) \to \exists Y: \phi(Y) \land \forall Z [\phi(Z) \to Z \not \prec Y], \text { for every formula } \phi$

Where: $X \approx Y \iff \forall z \, (z \in X \leftrightarrow z \in Y)$.

Let $\sf MKCC'$ be the theory $\sf MKCC$ with axiom of extensionality replaced by an axiom of extensionality for sets only.

Is $\sf MKCC'$ bi-interpretable with $\sf MKCC$?

$\endgroup$

1 Answer 1

8
$\begingroup$

No, because the latter theories has parametrically definable automorphisms that swap two equivalent classes, but the former theory is definably rigid (no need for class choice). If the theories were bi-interpretable, then we could fix a copy of the model with two equivalent classes (having the same elements), and inside this model we could define a copy of MK, and inside that define a copy of the original model, but because the automorphism would have to fix the definable interpretation but swap the classes, it would ruin the bi-interpretability feature.

$\endgroup$

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.