en>MadScientistX11: Removed primary source tags as I've added 4 non primary sources that verify the info

2015-01-10T00:48:27Z

Removed primary source tags as I've added 4 non primary sources that verify the info

en>AnomieBOT: Dating maintenance tags: {{Primary source inline}}

2014-01-29T15:55:37Z

Dating maintenance tags: {{Primary source inline}}

← Older revision		Revision as of 17:55, 29 January 2014
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	~~Obtaining an offshore account~~ is ~~one the best ways to get~~ a ~~person~~ to ~~protect their money,~~ and ~~something~~ of the ~~greatest kinds~~ of ~~offshore accounts are named no ID offshore account~~. ~~This type of account doesn't demand~~ a ~~person to confirm their identification when starting the account~~, as ~~well as~~ in ~~many situations~~ it~~'s properly fine for~~ a ~~person to open~~ the ~~account under a name~~ that is ~~diverse from their particular~~. ~~There are certainly~~ a ~~handful~~ of ~~points~~ a ~~person should know a few zero ID offshore account like~~ the ~~means~~ of ~~starting~~ a ~~merchant account~~, and ~~what most~~ of ~~these accounts usually are used for~~.<br><br>~~How exactly to open a no ID account~~<br><br>The ~~method~~ of ~~opening up~~ a ~~number identification account~~ is ~~extremely effortless~~. ~~Generally an individual can merely walk into a bank office~~, and then go through ~~the procedure~~ for ~~opening~~-up a ~~merchant account~~. ~~This usually involves having someone writedown the name of the account~~, and then ~~picking out what options they desire~~ for that ~~account. An individual can then just withdraw money from the account my writing down~~ a ~~key rule. Some banks offering offshore accounts will not possibly~~ need ~~a person~~ to ~~visit~~ the ~~financial institution workplace~~.<br><br>~~What these accounts are utilized for~~<br><br>~~These kind~~ of ~~accounts are often used~~ by ~~individuals who would like to protect their finances from things~~ such ~~as lawsuits~~, ~~or having~~ them ~~removed by way~~ of a ~~government~~. ~~This really~~ is ~~important~~, ~~since it~~ is ~~one~~ of ~~many techniques an individual~~ can ~~make certain that their income just does not get taken~~ for ~~just about any reason~~. ~~Like~~ [~~http~~:~~//anonymousprepaidcard1~~.~~tripod~~.~~com/~~ [~~http~~:/~~/anonymousprepaidcard1~~.~~tripod~~.~~com/ how do offshore accounts work~~]].		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).

			A [[cardinal number]] κ is called ''Mahlo'' if κ is [[inaccessible cardinal\|inaccessible]] and the [[Set (mathematics)\|set]] U = {λ < 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]]

en>A3 nm: /* OWL Full */ rewrite "It is unlikely that any reasoning software will be able to support complete reasoning for OWL Full." more strongly since OWL Full is known to be undecidable

2012-08-10T14:53:11Z

OWL Full: rewrite "It is unlikely that any reasoning software will be able to support complete reasoning for OWL Full." more strongly since OWL Full is known to be undecidable

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en>MadScientistX11: Removed primary source tags as I've added 4 non primary sources that verify the info

en>AnomieBOT: Dating maintenance tags: {{Primary source inline}}

en>A3 nm: /* OWL Full */ rewrite "It is unlikely that any reasoning software will be able to support complete reasoning for OWL Full." more strongly since OWL Full is known to be undecidable