Large Cardinal Axioms

• Strengthen ZFC by asserting existence of large infinite sets.
• Not provable in ZFC.

Examples:

• Inaccessible: $\kappa \text{ regular and }\forall \lambda <\kappa ,\mathcal{P}(\lambda )\subset V_{\kappa }$
• Measurable: $\exists \kappa \exists U\subset \mathcal{P}(\kappa )\text{ ultrafilter}$
• Mahlo, Weakly compact, Supercompact, Huge…