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{{no footnotes|date=November 2013}}
In [[set theory]], a branch of [[mathematics]], a '''rank-into-rank''' is a [[large cardinal]] λ satisfying one of the following four [[axiom]]s (commonly known as rank-into-rank embeddings, given in order of increasing consistency strength):
 
*'''Axiom I3:'''  There is a nontrivial [[elementary embedding]] of V<sub>λ</sub> into itself.
*'''Axiom I2:''' There is a nontrivial elementary embedding of V into a transitive class M that includes V<sub>λ</sub> where λ is the first fixed point above the [[critical point (set theory)|critical point]].
*'''Axiom I1:''' There is a nontrivial elementary embedding of V<sub>λ+1</sub> into itself.
*'''Axiom I0:'''  There is a nontrivial elementary embedding of L(V<sub>λ+1</sub>) into itself with the critical point below λ.
 
These are essentially the strongest known large cardinal axioms not known to be inconsistent in [[ZFC]]; the axiom for [[Reinhardt cardinal]]s is stronger, but is not consistent with the [[axiom of choice]].
 
If j is the elementary embedding mentioned in one of these axioms and κ is its [[critical point (set theory)|critical point]], then λ is the limit of <math>j^n(\kappa)</math> as n goes to ω. More generally, if the [[axiom of choice]] holds, it is provable that if there is a nontrivial elementary embedding of V<sub>α</sub> into itself then α is either a [[limit ordinal]] of [[cofinality]] ω or the successor of such an ordinal.
 
The axioms I1, I2, and I3 were at first suspected to be inconsistent (in ZFC) as it was thought possible that [[Kunen's inconsistency theorem]] that [[Reinhardt cardinal]]s are inconsistent with the axiom of choice could be extended to them, but this has not yet happened and they are now usually believed to be consistent.
 
Every I0 cardinal κ (speaking here of the critical point of ''j'') is an I1 cardinal.
 
Every I1 cardinal κ is an I2 cardinal and has a stationary set of I2 cardinals below it.
 
Every I2 cardinal κ is an I3 cardinal and has a stationary set of I3 cardinals below it.
 
Every I3 cardinal κ has another I3 cardinal ''above'' it and is an ''n''-[[huge cardinal]] for every ''n''<ω.
 
Axiom I1 implies that V<sub>λ+1</sub> (equivalently, H(λ<sup>+</sup>)) does not satisfy V=HOD.  There is no set S⊂λ definable in V<sub>λ+1</sub> (even from parameters V<sub>λ</sub> and ordinals &lt;λ<sup>+</sup>) with S cofinal in λ and |S|<λ, that is, no such S witnesses that λ is singular. And similarly for Axiom I0 and ordinal definability in L(V<sub>λ+1</sub>) (even from parameters in V<sub>λ</sub>). However globally, and even in V<sub>λ</sub>,<ref>Consistency of V = HOD With the Wholeness Axiom, Paul Corazza, Archive for Mathematical Logic, No. 39, 2000.</ref> V=HOD is relatively consistent with Axiom I1.
 
==References==
*{{citation|mr=0376347
|last=Gaifman|first= Haim
|chapter=Elementary embeddings of models of set-theory and certain subtheories|title=Axiomatic set theory |series=Proc. Sympos. Pure Math.|volume= XIII, Part II|pages= 33–101|publisher= Amer. Math. Soc.|publication-place=Providence R.I.|year= 1974}}
* {{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}}
* {{citation|last=Solovay|first=Robert M.|first2=William N. |last2=Reinhardt|first3= Akihiro |last3=Kanamori|year=1978|title=Strong axioms of infinity and elementary embeddings|journal=Annals of Mathematical Logic|volume=13|issue=1|pages=73–116|authorlink=Robert M. Solovay|doi=10.1016/0003-4843(78)90031-1 }}
{{Reflist}}
 
[[Category:Large cardinals]]
[[Category:Determinacy]]
 
 
{{settheory-stub}}

Latest revision as of 21:52, 6 January 2014

Template:No footnotes In set theory, a branch of mathematics, a rank-into-rank is a large cardinal λ satisfying one of the following four axioms (commonly known as rank-into-rank embeddings, given in order of increasing consistency strength):

  • Axiom I3: There is a nontrivial elementary embedding of Vλ into itself.
  • Axiom I2: There is a nontrivial elementary embedding of V into a transitive class M that includes Vλ where λ is the first fixed point above the critical point.
  • Axiom I1: There is a nontrivial elementary embedding of Vλ+1 into itself.
  • Axiom I0: There is a nontrivial elementary embedding of L(Vλ+1) into itself with the critical point below λ.

These are essentially the strongest known large cardinal axioms not known to be inconsistent in ZFC; the axiom for Reinhardt cardinals is stronger, but is not consistent with the axiom of choice.

If j is the elementary embedding mentioned in one of these axioms and κ is its critical point, then λ is the limit of as n goes to ω. More generally, if the axiom of choice holds, it is provable that if there is a nontrivial elementary embedding of Vα into itself then α is either a limit ordinal of cofinality ω or the successor of such an ordinal.

The axioms I1, I2, and I3 were at first suspected to be inconsistent (in ZFC) as it was thought possible that Kunen's inconsistency theorem that Reinhardt cardinals are inconsistent with the axiom of choice could be extended to them, but this has not yet happened and they are now usually believed to be consistent.

Every I0 cardinal κ (speaking here of the critical point of j) is an I1 cardinal.

Every I1 cardinal κ is an I2 cardinal and has a stationary set of I2 cardinals below it.

Every I2 cardinal κ is an I3 cardinal and has a stationary set of I3 cardinals below it.

Every I3 cardinal κ has another I3 cardinal above it and is an n-huge cardinal for every n<ω.

Axiom I1 implies that Vλ+1 (equivalently, H(λ+)) does not satisfy V=HOD. There is no set S⊂λ definable in Vλ+1 (even from parameters Vλ and ordinals <λ+) with S cofinal in λ and |S|<λ, that is, no such S witnesses that λ is singular. And similarly for Axiom I0 and ordinal definability in L(Vλ+1) (even from parameters in Vλ). However globally, and even in Vλ,[1] V=HOD is relatively consistent with Axiom I1.

References

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    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

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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.

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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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Template:Settheory-stub

  1. Consistency of V = HOD With the Wholeness Axiom, Paul Corazza, Archive for Mathematical Logic, No. 39, 2000.