Editing Extendible cardinal

In [[mathematics]], '''extendible cardinals''' are [[large cardinal]]s introduced by {{harvtxt|Reinhardt|1974}}, who was partly motivated by [[reflection principle]]s.  Intuitively, such a cardinal represents a point beyond which initial pieces of the [[Von Neumann universe|universe of sets]] start to look similar, in the sense that each is [[elementary embedding|elementarily embeddable]] into a later one.

==Definition==
For every [[ordinal number|ordinal]] ''η'', a [[cardinal number|cardinal]] κ is called '''η-extendible''' if  for some ordinal ''λ'' there is a nontrivial [[elementary embedding]] ''j'' of ''V''<sub>κ+η</sub> into ''V''<sub>λ</sub>, where ''κ'' is the [[critical point (set theory)|critical point]] of ''j'', and as usual ''V<sub>α</sub>'' denotes the ''α''th level of the [[Von Neumann universe|von Neumann hierarchy]].  A cardinal ''κ'' is called an '''extendible cardinal''' if it is ''η''-extendible for every nonzero ordinal ''η'' (Kanamori 2003).

==Properties==
For a cardinal <math>\kappa</math>, say that a logic <math>L</math> is <math>\kappa</math>-compact if for every set <math>A</math> of <math>L</math>-sentences, if every subset of <math>A</math> or cardinality <math><\kappa</math> has a model, then <math>A</math> has a model. (The usual [[compactness theorem]] shows <math>\aleph_0</math>-compactness of first-order logic.) Let <math>L_\kappa^2</math> be the [[infinitary logic]] for second-order set theory, permitting infinitary conjunctions and disjunctions of length <math><\kappa</math>. <math>\kappa</math> is extendible iff <math>L_\kappa^2</math> is <math>\kappa</math>-compact.<ref>{{cite journal
| last1=Magidor | first1=M. | authorlink1=Menachem Magidor
| title=On the Role of Supercompact and Extendible Cardinals in Logic
| date=1971
| pages=147–157
| journal=[[Israel Journal of Mathematics]]
| volume=10
| issue=2
| doi=10.1007/BF02771565 | doi-access=free}}</ref>

==Variants and relation to other cardinals==
A cardinal ''κ'' is called ''η-C<sup>(n)</sup>''-extendible if there is an elementary embedding ''j'' witnessing that ''κ'' is ''η''-extendible (that is, ''j'' is elementary from ''V<sub>κ+η</sub>'' to some ''V<sub>λ</sub>'' with critical point ''κ'') such that furthermore, ''V<sub>j(κ)</sub>'' is ''Σ<sub>n</sub>''-correct in ''V''.  That is, for every [[Lévy hierarchy#Definitions|''Σ<sub>n</sub>'']] formula ''φ'', ''φ'' holds in ''V<sub>j(κ)</sub>'' if and only if ''φ'' holds in ''V''.  A cardinal ''κ'' is said to be '''C<sup>(n)</sup>-extendible''' if it is ''η-C<sup>(n)</sup>''-extendible for every ordinal ''η''.  Every extendible cardinal is ''C<sup>(1)</sup>''-extendible, but for ''n≥1'', the least ''C<sup>(n)</sup>''-extendible cardinal is never ''C<sup>(n+1)</sup>''-extendible (Bagaria 2011).

[[Vopěnka's principle]] implies the existence of extendible cardinals; in fact, Vopěnka's principle (for definable classes) is equivalent to the existence of ''C<sup>(n)</sup>''-extendible cardinals for all ''n'' (Bagaria 2011).  All extendible cardinals are [[supercompact cardinal]]s (Kanamori 2003).

==See also==
*[[List of large cardinal properties]]
*[[Reinhardt cardinal]]

==References==
{{reflist}}
{{refbegin}}
*{{cite journal|last1=Bagaria|first1=Joan|title=''C<sup>(n)</sup>''-cardinals|journal=Archive for Mathematical Logic|date=23 December 2011|volume=51|issue=3–4|pages=213–240|doi=10.1007/s00153-011-0261-8|s2cid=208867731 }}
*{{cite web|last1=Friedman|first1=Harvey|authorlink=Harvey Friedman (mathematician)|title=Restrictions and Extensions|url=http://u.osu.edu/friedman.8/files/2014/01/ResExt021703-1t4vsx4.pdf}}
*{{cite book|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=0401475
|authorlink=William Nelson Reinhardt|last=Reinhardt|first= W. N.
|chapter=Remarks on reflection principles, large cardinals, and elementary embeddings. |title=Axiomatic set theory |series=Proc. Sympos. Pure Math.|volume= XIII, Part II|pages= 189–205|publisher= Amer. Math. Soc.|publication-place= Providence, R. I.|year= 1974}}
{{refend}}

[[Category:Large cardinals]]


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