Mahlo cardinal: Difference between revisions

"Ord is Mahlo" is an assertion that, as one can likely guess, asserts that every function \( f: \mathrm{Ord} \to \mathrm{Ord} \) has a strongly inaccessible closure point. Clearly, "Ord is Mahlo" implies that there is a proper class of inaccessible cardinals, 1-inaccessible cardinals, and more. However, if \( \kappa \) is Mahlo, then \( V_\kappa \) satisfies "Ord is Mahlo", and thus "Ord is Mahlo" has consistency strength squashed between the inaccessible hierarchy and strongly Mahlo cardinals.

 

 

Furthermore, let \( \mathrm{MP}(\mathbb{R}) \), the maximality principle for the real numbers be the following statement: "assume \( r \) is a real number and \( \varphi \) is a formula. Then if there is a forcing extension \( V[G] \) so that \( \varphi(r) \) and \( \varphi(r) \) persists, i.e. remains true in all subsequent extensions \( V[G][H] \), then \( \varphi(r) \) is already true in the universe". Essentially, the theory of the real numbers is already maximal, and it's not possible to persistently force a statement that isn't true to be true. The statement \( \mathrm{MP}(\mathbb{R}) \) has less consistency strength than \( \mathrm{MP}(V) \), where \( r \) is an arbitrary set, and is actually equiconsistent with Ord is Mahlo.