Editing Rank-into-rank

{{more footnotes|date=November 2013}} 
In [[set theory]], a branch of [[mathematics]], a '''rank-into-rank''' embedding is a [[large cardinal property]] defined by one of the following four [[axiom]]s given in order of increasing consistency strength. (A set of rank <math>< \lambda</math> is one of the elements of the set <math>V_\lambda</math> of the [[von Neumann hierarchy]].)

*'''Axiom I3:'''  There is a nontrivial [[elementary embedding]] of <math>V_\lambda</math> into itself.
*'''Axiom I2:''' There is a nontrivial elementary embedding of <math>V</math> into a transitive class <math>M</math> that includes <math>V_\lambda</math> where <math>\lambda</math> is the first fixed point above the [[critical point (set theory)|critical point]].
*'''Axiom I1:'''  There is a nontrivial elementary embedding of <math>V_{\lambda+1}</math> into itself.
*'''Axiom I0:'''  There is a nontrivial elementary embedding of <math>L(V_{\lambda+1})</math> into itself with critical point below <math>\lambda</math>.

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 <math>j</math> is the elementary embedding mentioned in one of these axioms and <math>\kappa</math> is its [[critical point (set theory)|critical point]], then <math>\lambda</math> is the limit of <math>j^n(\kappa)</math> as <math>n</math> goes to <math>\omega</math>.  More generally, if the [[axiom of choice]] holds, it is provable that if there is a nontrivial elementary embedding of <math>V_\alpha</math> into itself then <math>\alpha</math> is either a [[limit ordinal]] of [[cofinality]] <math>\omega</math> or the successor of such an ordinal.

The axioms I0, 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 <math>\kappa</math> (speaking here of the critical point of <math>j</math>) is an I1 cardinal.

Every I1 cardinal <math>\kappa</math> (sometimes called ω-huge cardinals) is an I2 cardinal and has a stationary set of I2 cardinals below it.

Every I2 cardinal <math>\kappa</math> is an I3 cardinal and has a stationary set of I3 cardinals below it.

Every I3 cardinal <math>\kappa</math> has another I3 cardinal ''above'' it and is an <math>n</math>-[[huge cardinal]] for every <math>n< \omega</math>.

Axiom I1 implies that <math>V_{\lambda+1}</math> (equivalently, <math>H(\lambda^+)</math>) does not satisfy V=[[hereditarily ordinal definable|HOD]].  There is no set <math>S\subset\lambda</math> definable in <math>V_{\lambda+1}</math> (even from parameters <math>V_\lambda</math> and ordinals <math><\lambda^+</math>) with <math>S</math> cofinal in <math>\lambda</math> and <math>\vert S\vert<\lambda</math>, that is, no such <math>S</math> witnesses that <math>\lambda</math> is singular. And similarly for Axiom I0 and ordinal definability in <math>L(V_{\lambda+1})</math> (even from parameters in <math>V_\lambda</math>). However globally, and even in <math>V_\lambda</math>,<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.

Notice that I0 is sometimes strengthened further by adding an "Icarus set", so that it would be
*'''Axiom Icarus set:'''  There is a nontrivial elementary embedding of <math>L(V_{\lambda+1}, \mathrm{Icarus})</math> into itself with the critical point below <math>\lambda</math>.
The Icarus set should be in <math>V_{\lambda+1}\setminus L(V_{\lambda+1})</math> but chosen to avoid creating an inconsistency. So for example, it cannot encode a well-ordering of <math>V_{\lambda+1}</math>. See section 10 of Dimonte for more details.

Woodin defined a sequence of sets <math>E_\alpha(V_{\lambda+1})</math> for use as Icarus sets.<ref>V. Dimonte, "[https://link.springer.com/article/10.1007/s00153-011-0232-0 Totally non-proper ordinals beyond <math>L(V_{\lambda+1})</math>]". Archive for Mathematical Logic vol. 50 (2011), p.570--571. (Available at "[https://typeset.io/pdf/totally-non-proper-ordinals-beyond-l-v-l-1-46a913ia2s.pdf typeset.io]", pp.8--9.)</ref>

==Notes==
{{reflist}}

==References==
*{{cite arXiv
 |last=Dimonte |first= Vincenzo 
 |date=2017
 |title=I0 and rank-into-rank axioms
 |eprint=1707.02613
 |mode=cs2
|class= math.LO 
 }}.
*{{citation|mr=0376347
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* {{Citation|last=Kanamori|first=Akihiro|authorlink=Akihiro Kanamori|year=2003|publisher=Springer|title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings|title-link=The Higher Infinite|edition=2nd|isbn=3-540-00384-3}}.
*{{citation|mr=1489305 
|last=Laver|first= Richard|authorlink=Richard Laver
|title=Implications between strong large cardinal axioms
|journal=Ann. Pure Appl. Logic|volume= 90 |year=1997|issue= 1–3|pages= 79–90|doi=10.1016/S0168-0072(97)00031-6|doi-access=free}}. 
* {{citation|last1=Solovay|first1=Robert M.|authorlink2=William Nelson Reinhardt|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|doi-access=free}}.

[[Category:Large cardinals]]
[[Category:Determinacy]]