Editing Mahlo cardinal (section)

==Mahlo cardinals and reflection principles==

Axiom F is the statement that every normal function on the ordinals has a regular fixed point. (This is not a first-order axiom as it quantifies over all normal functions, so it can be considered either as a second-order axiom or as an axiom scheme.)
A cardinal is called Mahlo if every normal function on it has a regular fixed point{{citation needed|date=December 2022}}, so axiom F is in some sense saying that the class of all ordinals is Mahlo.{{citation needed|date=July 2023}} A cardinal κ is Mahlo if and only if a second-order form of axiom F holds in ''V''<sub>κ</sub>.{{citation needed|date=July 2023}} Axiom F is in turn equivalent to the statement that for any formula φ with parameters there are arbitrarily large inaccessible ordinals α such that ''V''<sub>α</sub> reflects φ (in other words φ holds in ''V''<sub>α</sub> if and only if it holds in the whole universe) {{harv|Drake|1974|loc=chapter 4}}.