Mahlo cardinal: Difference between revisions

In [[mathematics]], a '''Mahlo cardinal''' is a certain kind of [[large cardinal]] number. Mahlo cardinals were first described by {{harvs|txt|authorlink= Paul Mahlo|first=Paul|last=Mahlo|year1=1911|year2=1912|year3=1913}}. As with all large cardinals, none of these varieties of Mahlo cardinals can be provedproven to exist by [[ZFC]] (assuming ZFC is [[consistent theory|consistent]]).

A [[cardinal number]] κ<math>\kappa</math> is called '''strongly Mahlo''' if κ<math>\kappa</math> is [[inaccessible cardinal|strongly inaccessible]] and the [[Set (mathematics)|set]] <math>U = \{λ\lambda<\kappa &lt;\mid κ: λ\lambda\text{ is strongly inaccessible}\}</math> is [[stationary set#Classical notion|stationary]] in κ.

A cardinal κ<math>\kappa</math> is called '''weakly Mahlo''' if κ<math>\kappa</math> is weakly inaccessible and the set of weakly inaccessible cardinals less than κ<math>\kappa</math> is stationary in κ<math>\kappa</math>.

== Example: showing that Mahlo cardinals κ are hyperκ-inaccessible (hyper-inaccessible)==

Aa cardinal κ is called α-Mahlo for some ordinal α if and only if κ is Mahlostrongly inaccessible and for every ordinal β<α, the set of β-Mahlo cardinals below κ is stationary in κ.<ref>W. Boos, ''[https://www.proquest.com/openview/22276484b3832b4d487425ee882b5b09/1 Nonstandard Large Cardinals]'' (1971).</ref>^{p. 3} However the condition "κ is strongly inaccessible" is sometimes replaced by other conditions, such as "κ is regular" or "κ is weakly inaccessible" or "κ is Mahlo". 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''' or '''κ⁺-Mahlo''' if and only if it is inaccessible and there is a normal (i.e. nontrivial and closed under [[diagonal intersection]]s) κ-complete [[Filter (mathematics)|filter]] on the power set of κ that is closed under the Mahlo operation, which maps the set of ordinals ''S'' to {α<math>\in</math>''S'': α has uncountable cofinality and S∩α is stationary in α}

If ''X'' is a class of ordinals, themthen we can form a new class of ordinals ''M''(''X'') consisting of the ordinals α of uncountable cofinality such that α∩''X'' is stationary in α. This operation ''M'' is called the '''Mahlo operation'''. It can be used to define Mahlo cardinals: for example, if ''X'' is the class of regular cardinals, then ''M''(''X'') is the class of weakly Mahlo cardinals. 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.

AAxiom cardinalF is called Mahlo if every normal function on it has a fixed point, so the axiom (scheme) statingstatement that "every normal function on the ordinals has a regular fixed point" is in some sense saying that the class of all ordinals is Mahlo. (This axiom is innot turna equivalentfirst-order toaxiom theas statementit thatquantifies thereover areall arbitrarilynormal largefunctions, inaccessibleso ordinalsit αcan suchbe thatconsidered ''V''_αeither isas ana elementarysecond-order submodelaxiom ofor theas universean {{harv|Drake|1974|loc=chapteraxiom 4}}scheme.)