Математикада Махло кардиналдары

Concept in mathematics
In mathematics, a Mahlo cardinal is a certain kind of large cardinal number. Mahlo cardinals were first described by As with all large cardinals, none of these varieties of Mahlo cardinals can be proven to exist by ZFC (assuming ZFC is consistent). A cardinal number is called strongly Mahlo if is strongly inaccessible and the set is stationary in κ. A cardinal is called weakly Mahlo if is weakly inaccessible and the set of weakly inaccessible cardinals less than is stationary in
The term "Mahlo cardinal" now usually means "strongly Mahlo cardinal", though the cardinals originally considered by Mahlo were weakly Mahlo cardinals.

If κ is a limit ordinal and the set of regular ordinals less than κ is stationary in κ, then κ is weakly Mahlo. The main difficulty in proving this is to show that κ is regular. We will suppose that it is not regular and construct a club set which gives us a μ such that:
μ = cf(μ) < cf(κ) < μ < κ which is a contradiction. If κ were not regular, then cf(κ) < κ. We could choose a strictly increasing and continuous cf(κ) sequence which begins with cf(κ)+1 and has κ as its limit. The limits of that sequence would be club in κ. So there must be a regular μ among those limits. So μ is a limit of an initial subsequence of the cf(κ) sequence. Thus its cofinality is less than the cofinality of κ and greater than it at the same time; which is a contradiction. Thus the assumption that κ is not regular must be false, i. e. κ is regular. No stationary set can exist below with the required property because {2,3,4, } is club in ω but contains no regular ordinals; so κ is uncountable. And it is a regular limit of regular cardinals; so it is weakly inaccessible. Then one uses the set of uncountable limit cardinals below κ as a club set to show that the stationary set may be assumed to consist of weak inaccessibles. If κ is weakly Mahlo and also a strong limit, then κ is Mahlo. κ is weakly inaccessible and a strong limit, so it is strongly inaccessible. We show that the set of uncountable strong limit cardinals below κ is club in κ. Let μ0 be the larger of the threshold and ω1. For each finite n, let μn+1 = 2μn which is less than κ because it is a strong limit cardinal. Then their limit is a strong limit cardinal and is less than κ by its regularity. The limits of uncountable strong limit cardinals are also uncountable strong limit cardinals. So the set of them is club in κ. Intersect that club set with the stationary set of weakly inaccessible cardinals less than κ to get a stationary set of strongly inaccessible cardinals less than κ.

The term "hyper inaccessible" is ambiguous. 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). Suppose κ is Mahlo. We proceed by transfinite induction on α to show that κ is α inaccessible for any α ≤ κ. Since κ is Mahlo, κ is inaccessible; and thus 0 inaccessible, which is the same thing. If κ is α inaccessible, then there are β inaccessibles (for β < α) arbitrarily close to κ. 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)). It is closed, so it is club in κ. So, by κ's Mahlo ness, it contains an inaccessible. That inaccessible is actually an α inaccessible. So κ is α+1 inaccessible. If λ ≤ κ is a limit ordinal and κ is α inaccessible for all α < λ, then every β < λ is also less than α for some α < λ. So this case is trivial. In particular, κ is κ inaccessible and thus hyper inaccessible. 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. Choose a 0 inaccessible above the threshold, call it α0. Then pick an α0 inaccessible, call it α1. Keep repeating this and taking limits at limits until you reach a fixed point, call it μ. 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. So κ is 1 hyper inaccessible. We can intersect this same club set with the stationary set less than κ to get a stationary set of hyper inaccessibles less than κ. The rest of the proof that κ is α hyper inaccessible mimics the proof that it is α inaccessible. So κ is hyper hyper inaccessible, etc

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, so axiom F is in some sense saying that the class of all ordinals is Mahlo. A cardinal κ is Mahlo if and only if a second order form of axiom F holds in Vκ. 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) .

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. Let be , the fold iterated Cartesian product of the closed unit interval with itself. The group of all permutations of that move only finitely many natural numbers can be seen as acting on by permuting coordinates. The group action also acts diagonally on any of the products , by defining an abuse of notation For , let if and are in the same orbit under this diagonal action. Let be a Borel function such that for any and , if then Then there is a sequence such that for all sequences of indices , is the first coordinate of This theorem is provable in , but not in any theory for some fixed .