ZFC is a common axiomatic system, and so is MK. The latter is significantly stronger since it can deal with proper classes.
Definition: A cardinal $\kappa$ is inaccessible if it is regular and the regular cardinals less than it are unbounded in it.
Theorem: For any inaccessible cardinal $\kappa$, $(V_\kappa, \in) \vDash \text{ZFC}$, and $(V_\kappa, V_{\kappa+1}, \in) \vDash \text{MK}$.
Because of this theorem, I was wondering which of the following theories is stronger:
- $\text{ZFC}$ augmented by the axiom "there exists an inaccessible cardinal" (I denote this $\text{ZFCi}$).
- $\text{ZFC}$ augmented by the axiom "MK set theory is consistent" (I denote this $\text{ZFCK}$).
In other words, I was wondering which of the following, if any, holds:
- $\text{ZFCi} \vDash \text{Con(ZFCK)}$
- $\text{ZFCK} \vDash \text{Con(ZFCi)}$
$\mathsf{ZFCi}$ proves that there is a $\kappa$ such that $(V_{\kappa+1},V_\kappa,\in)\models\mathsf{MK}$. Consequently, $\mathsf{ZFCi}\vdash \mathsf{Con}(\mathsf{MK})$, which in turn means that $\mathsf{ZFCi}\vdash$ "Every $\omega$-model satisfies $\mathsf{Con(MK)}$" since consistency is arithmetical. Since $V_\kappa$ is an $\omega$-model, this means $\mathsf{ZFCi}\vdash \exists \kappa((V_\kappa,\in)\models\mathsf{ZFCK})$.
Godel's second incompleteness theorem then tells us that $\mathsf{ZFCK}\not\vdash\mathsf{Con(ZFCi)}$.