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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 proved to exist by [[ZFC]] (assuming ZFC is consistent).
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A [[cardinal number]] κ is called ''Mahlo'' if κ is [[inaccessible cardinal|inaccessible]] and the [[Set (mathematics)|set]] U = {λ &lt; κ: λ is inaccessible} is [[stationary set#Classical notion|stationary]] in κ.
 
A cardinal κ is called ''weakly Mahlo'' if  κ is weakly inaccessible and the set of weakly inaccessible cardinals less than κ is stationary in κ.
 
== Minimal condition sufficient for a Mahlo cardinal ==
 
* 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 <math>\aleph_0</math> 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 μ<sub>0</sub> be the larger of the threshold and ω<sub>1</sub>. For each finite n, let μ<sub>n+1</sub> = 2<sup>μ<sub>n</sub></sup> 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.
 
== Example: showing that Mahlo cardinals are hyper-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 [[inaccessible cardinal|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 α<sub>0</sub>. Then pick an α<sub>0</sub>-inaccessible, call it α<sub>1</sub>.  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..
 
== More than just Mahlo ==
 
A cardinal κ is α-Mahlo for some ordinal α if and only if κ is Mahlo and for every ordinal β<α, the set of β-Mahlo cardinals below κ is stationary in κ.  We can define "hyper-Mahlo", "α-hyper-Mahlo", "weakly α-Mahlo", "weakly hyper-Mahlo", "weakly α-hyper-Mahlo", etc. by analogy with the definitions for inaccessibles.
 
A cardinal κ is '''greatly Mahlo''' or '''κ<sup>+</sup>-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 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}
 
The properties of being inaccessible, Mahlo, weakly Mahlo, α-Mahlo, greatly Mahlo, etc. are preserved if we replace the universe by an [[inner model]].
 
==See also==
*[[Inaccessible cardinal]]
*[[Stationary set]]
*[[Inner model]]
 
== References ==
* {{cite book| last=Drake | first=Frank R. |title=Set Theory: An Introduction to Large Cardinals | series=Studies in Logic and the Foundations of Mathematics | volume=76)|publisher=Elsevier Science Ltd | year=1974 | isbn=0-444-10535-2 | zbl=0294.02034 }}
* {{cite book | last=Kanamori | first=Akihiro | authorlink=Akihiro Kanamori | year=2003 | publisher=[[Springer-Verlag]] | title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings | series=Springer Monographs in Mathematics | edition=2nd | isbn=3-540-00384-3 | zbl=1022.03033  }}
*{{Citation | last1=Mahlo | first1=Paul | authorlink=Paul Mahlo | title=Über lineare transfinite Mengen | year=1911 | journal=Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Mathematisch-Physische Klasse | volume=63 | pages=187–225 | zbl=42.0090.02}}
*{{Citation | last1=Mahlo | first1=Paul | authorlink=Paul Mahlo | title=Zur Theorie und Anwendung der ρ<sub>0</sub>-Zahlen | year=1912 | journal=Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Mathematisch-Physische Klasse | volume=64 | pages=108–112 | zbl=43.0113.01 }}
 
[[Category:Large cardinals]]

Latest revision as of 02:48, 10 January 2015

New York Town is among the most populous cities of the globe. It has a lot of influence in excess of world-wide finance, media, vogue, tradition, artwork, commerce, investigation, and enjoyment. A New York Town trip is the very best way to know the city and see its various points of interest Belinda Broido .

The Masonic Temple is one of the oldest buildings in Binghamton. It stands tall, a mere shadow of its previous glory. Broken home windows offer perches for pigeons. The walls draw the interest of graffitti vandals. The ornate columns shelter wayward learners evading classes in the college.

We realized of the law enforcement existence at the Masonic Temple at five:31pm. At that time there was an officer on guard and police tape blocking the entrance. There experienced currently been a sweep of the constructing to make sure neither vagrants nor cautious young children experienced ventured within. Also on scene was a fire marshall, with a letter prepared to wonderful the proprietors of the creating for the several code violations, prepared for the mail. The Parks Dept was en route to seal the constructing when once again.

It's tough to explain how it feels to be the only American completely off the earth at a time such as this. The emotion that I need to be there with all of you, working with this, assisting in some way, is mind-boggling. I know that we are on the threshold (or beyond) of a terrible shift in the heritage of the world. Numerous things will never be the very same yet again soon after September 11, 2001. Not just for the thousands and countless numbers of men and women straight impacted by these horrendous acts of terrorism, but possibly for all of us. We will locate ourselves feeling otherwise about dozens of things, like most likely space exploration, sadly.

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Another museum in the metropolis is the Museum of Contemporary Artwork which is regarded one particular of the greatest contemporary art museums in the globe. It has become far better right after its latest renewal by Yoshio Taniguchi. This building was elevated below the supervision of Edward Durrell and Philip Goodwin. The museum's galleries contain a lot of well-known prints, drawings, illustrations, photographs and paintings.

Coarse and/or Nappy: Any black female that has been in Belinda Broido lengthier than two months is aware that if you want to get your hair braided then one hundred and twenty fifth avenue is the epicenter. We are continuously stopped by girls sporting classic African clothes, asking "do you want your hair braided.' If you ever consider one of them up on that offer you, you may possibly discover by yourself stuffed in again of a retailer front with fifty other folks obtaining large box braids, hunting like Janet Jackson's Justice circa 1993, when you asked for micro-braids. Rather when you see a lady on the teach with braids that you like, request her where she went and if she suggests one hundred and twenty fifth request her for specifics. Performing your homework will genuinely pay off and this will help save you the headache of making an attempt to go at it by itself.

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