As with all large cardinals, none of these varieties of Mahlo cardinals can be proven to exist by ZFC (assuming ZFC is consistent).
is strongly inaccessible, and for any unbounded set
The main difficulty in proving this is to show that κ is regular.
So μ is a limit of an initial subsequence of the cf(κ)-sequence.
with the required property because {2,3,4,...} is club in ω but contains no regular ordinals; so κ is uncountable.
For each finite n, let μn+1 = 2μn which is less than κ because it is a strong limit cardinal.
In this section, a cardinal κ is called hyper-inaccessible if it is κ-inaccessible (as opposed to the more common meaning of 1-inaccessible).
Consider the set of simultaneous limits of such β-inaccessibles larger than some threshold but less than κ.
It is unbounded in κ (imagine rotating through β-inaccessibles for β < α ω-times choosing a larger cardinal each time, then take the limit which is less than κ by regularity (this is what fails if α ≥ κ)).
To show that κ is a limit of hyper-inaccessibles and thus 1-hyper-inaccessible, we need to show that the diagonal set of cardinals μ < κ which are α-inaccessible for every α < μ is club in κ.
Then μ has the required property (being a simultaneous limit of α-inaccessibles for all α < μ) and is less than κ by regularity.
Limits of such cardinals also have the property, so the set of them is club in κ.
By Mahlo-ness of κ, there is an inaccessible in this set and it is hyper-inaccessible.
The term α-Mahlo is ambiguous and different authors give inequivalent definitions.
We can define "hyper-Mahlo", "α-hyper-Mahlo", "hyper-hyper-Mahlo", "weakly α-Mahlo", "weakly hyper-Mahlo", "weakly α-hyper-Mahlo", and so on, by analogy with the definitions for inaccessibles, so for example a cardinal κ is called hyper-Mahlo if it is κ-Mahlo.
A regular uncountable cardinal κ is greatly Mahlo if and only if there is a normal (i.e. nontrivial and closed under diagonal intersections) κ-complete filter on the power set of κ that is closed under the Mahlo operation, which maps the set of ordinals S to {α
S: α has uncountable cofinality and S∩α is stationary in α} For α < κ+, define the subsets Mα(κ) ⊆ κ inductively as follows: Although the exact definition depends on a choice of cofinal subset for each α < κ+ of cofinality κ, any choice will give the same sequence of subsets modulo the nonstationary ideal.
Every reflecting cardinal has strictly more consistency strength than a greatly Mahlo, but inaccessible reflecting cardinals aren't in general Mahlo.
The condition that α has uncountable cofinality ensures that the closed unbounded subsets of α are closed under intersection and so form a filter; in practice the elements of X often already have uncountable cofinality in which case this condition is redundant.
Some authors add the condition that α is in X, which in practice usually makes little difference as it is often automatically satisfied.
For a fixed regular uncountable cardinal κ, the Mahlo operation induces an operation on the Boolean algebra of all subsets of κ modulo the non-stationary ideal.
The Mahlo operation can be iterated transfinitely as follows: These iterated Mahlo operations produce the classes of α-Mahlo cardinals starting with the class of strongly inaccessible cardinals.
The diagonalized Mahlo operation produces the hyper-Mahlo cardinals, and so on.
Axiom F is the statement that every normal function on the ordinals has a regular fixed point.
A cardinal is called Mahlo if every normal function on it has a regular fixed point[citation needed], so axiom F is in some sense saying that the class of all ordinals is Mahlo.
[citation needed] A cardinal κ is Mahlo if and only if a second-order form of axiom F holds in Vκ.
[citation needed] 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 Vα reflects φ (in other words φ holds in Vα if and only if it holds in the whole universe) (Drake 1974, chapter 4).
Harvey Friedman (1981) has shown that existence of Mahlo cardinals is a necessary assumption in a sense to prove certain theorems about Borel functions on products of the closed unit interval.
-fold iterated Cartesian product of the closed unit interval with itself.
that move only finitely many natural numbers can be seen as acting on