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In [[mathematics]], a '''weakly compact cardinal''' is a certain kind of [[cardinal number]] introduced by {{harvtxt|Erdős|Tarski|1961}}; weakly compact cardinals are [[large cardinal]]s, meaning that their existence can not be proven  from the [[ZFC|standard axioms of set theory]].
 
Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function ''f'': [κ] <sup> 2 </sup> → {0, 1} there is a [[Set (mathematics)|set]] of [[cardinality]] κ that is [[Homogeneous (large cardinal property)|homogeneous]] for ''f''. In this context, [κ] <sup> 2 </sup> means the set of 2-element subsets of κ, and a subset ''S'' of κ is homogeneous for ''f'' [[if and only if]] either all of [''S'']<sup>2</sup> maps to 0 or all of it maps to 1.
 
The name "weakly compact" refers to the fact that if a cardinal is weakly compact then a certain related [[infinitary language]] satisfies a version of the compactness theorem; see below.  
 
Weakly compact cardinals are [[Mahlo cardinal]]s, and the set of Mahlo cardinals less than a given weakly compact cardinal is stationary.
 
Some authors use a weaker definition of weakly compact cardinals, such as one of the conditions below with the condition of inaccessibility dropped.
 
== Equivalent formulations ==
 
The following are equivalent for any [[uncountable]] cardinal κ:
 
# κ is weakly compact.
# for every λ<κ, natural number n ≥ 2, and function f: [κ]<sup>n</sup> → λ, there is a set of cardinality κ that is homogeneous for f. {{harv|Drake|1974|loc=chapter 7 theorem 3.5}}
# κ is [[Inaccessible cardinal|inaccessible]] and has the [[tree property]], that is, every [[Tree (set theory)|tree]] of height κ has either a level of size κ or a branch of size κ.
# Every linear order of cardinality κ has an ascending or a descending sequence of order type κ.
# κ is <math>\Pi^1_1</math>-[[totally indescribable cardinal|indescribable]].
# κ has the extension property. In other words, for all ''U'' ⊂ ''V''<sub>κ</sub> there exists a transitive set ''X'' with κ ∈ ''X'', and a subset ''S'' ⊂ ''X'', such that (''V''<sub>κ</sub>, , ''U'') is an [[elementary substructure]] of (''X'', ∈, ''S''). Here, ''U'' and ''S'' are regarded as unary [[Predicate (mathematical logic)|predicates]].
# For every set S of cardinality κ of subsets of κ, there is a non-trivial κ-complete filter that decides S.
# κ is κ-[[unfoldable cardinal|unfoldable]].
# κ is inaccessible and the [[infinitary language]] ''L''<sub>κ,κ</sub> satisfies the weak compactness theorem.  
# κ is inaccessible and the [[infinitary language]] ''L''<sub>κ,ω</sub> satisfies the weak compactness theorem.
 
A language ''L''<sub>κ,κ</sub> is said to satisfy the weak compactness theorem if whenever Σ is a set of sentences of cardinality at most κ and every subset with less than κ elements has a model, then Σ has a model. [[Strongly compact cardinal]]s are defined in a similar way without the restriction on the cardinality of the set of sentences.
 
==See also==
*[[list of large cardinal properties]]
 
== References ==
 
* {{citation|last=Drake|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|first =F. R.}}
*{{citation|authorlink1=Paul Erdős|authorlink2=Alfred Tarski|last= Erdős|first= Paul|last2=Tarski|first2= Alfred|chapter= On some problems involving inaccessible cardinals|year=  1961 |title= Essays on the foundations of mathematics  |pages= 50–82 |publisher=Magnes Press, Hebrew Univ.|publication-place= Jerusalem|url=http://www.renyi.hu/~p_erdos/|mr=0167422}}
* {{citation|last=Kanamori|first=Akihiro|authorlink=Akihiro Kanamori|year=2003|publisher=Springer|title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings|edition=2nd |isbn= 3-540-00384-3}}
 
[[Category:Large cardinals]]

Latest revision as of 19:01, 14 November 2013

In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by Template:Harvtxt; weakly compact cardinals are large cardinals, meaning that their existence can not be proven from the standard axioms of set theory.

Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function f: [κ] 2 → {0, 1} there is a set of cardinality κ that is homogeneous for f. In this context, [κ] 2 means the set of 2-element subsets of κ, and a subset S of κ is homogeneous for f if and only if either all of [S]2 maps to 0 or all of it maps to 1.

The name "weakly compact" refers to the fact that if a cardinal is weakly compact then a certain related infinitary language satisfies a version of the compactness theorem; see below.

Weakly compact cardinals are Mahlo cardinals, and the set of Mahlo cardinals less than a given weakly compact cardinal is stationary.

Some authors use a weaker definition of weakly compact cardinals, such as one of the conditions below with the condition of inaccessibility dropped.

Equivalent formulations

The following are equivalent for any uncountable cardinal κ:

  1. κ is weakly compact.
  2. for every λ<κ, natural number n ≥ 2, and function f: [κ]n → λ, there is a set of cardinality κ that is homogeneous for f. Template:Harv
  3. κ is inaccessible and has the tree property, that is, every tree of height κ has either a level of size κ or a branch of size κ.
  4. Every linear order of cardinality κ has an ascending or a descending sequence of order type κ.
  5. κ is -indescribable.
  6. κ has the extension property. In other words, for all UVκ there exists a transitive set X with κ ∈ X, and a subset SX, such that (Vκ, ∈, U) is an elementary substructure of (X, ∈, S). Here, U and S are regarded as unary predicates.
  7. For every set S of cardinality κ of subsets of κ, there is a non-trivial κ-complete filter that decides S.
  8. κ is κ-unfoldable.
  9. κ is inaccessible and the infinitary language Lκ,κ satisfies the weak compactness theorem.
  10. κ is inaccessible and the infinitary language Lκ,ω satisfies the weak compactness theorem.

A language Lκ,κ is said to satisfy the weak compactness theorem if whenever Σ is a set of sentences of cardinality at most κ and every subset with less than κ elements has a model, then Σ has a model. Strongly compact cardinals are defined in a similar way without the restriction on the cardinality of the set of sentences.

See also

References

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    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

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  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

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    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

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