Extendible cardinal#Variants and relation to other cardinals

For every ordinal η, a cardinal κ is called η-extendible if for some ordinal λ there is a nontrivial elementary embedding j of V_κ+η into V_λ, where κ is the critical point of j, and as usual V_α denotes the αth level of the von Neumann hierarchy. A cardinal κ is called an extendible cardinal if it is η-extendible for every nonzero ordinal η (Kanamori 2003).

For a cardinal $\backslash kappa$, say that a logic $L$ is $\backslash kappa$-compact if for every set $A$ of $L$-sentences, if every subset of $A$ or cardinality has a model, then $A$ has a model. (The usual compactness theorem shows $\backslash aleph\_0$-compactness of first-order logic.) Let $L\_\backslash kappa^2$ be the infinitary logic for second-order set theory, permitting infinitary conjunctions and disjunctions of length . $\backslash kappa$ is extendible iff $L\_\backslash kappa^2$ is $\backslash kappa$-compact.{{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}}

A cardinal κ is called η-C⁽ⁿ⁾-extendible if there is an elementary embedding j witnessing that κ is η-extendible (that is, j is elementary from V_κ+η to some V_λ with critical point κ) such that furthermore, V_j(κ) is Σ_n-correct in V. That is, for every Σ_n formula φ, φ holds in V_j(κ) if and only if φ holds in V. A cardinal κ is said to be C⁽ⁿ⁾-extendible if it is η-C⁽ⁿ⁾-extendible for every ordinal η. Every extendible cardinal is C⁽¹⁾-extendible, but for n≥1, the least C⁽ⁿ⁾-extendible cardinal is never C⁽ⁿ⁺¹⁾-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⁽ⁿ⁾-extendible cardinals for all n (Bagaria 2011). All extendible cardinals are supercompact cardinals (Kanamori 2003).

• List of large cardinal properties
• Reinhardt cardinal

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• {{cite journal|last1=Bagaria|first1=Joan|title=C⁽ⁿ⁾-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}}

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Category:Large cardinals

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