Extendible cardinal explained

In mathematics, extendible cardinals are large cardinals introduced by, who was partly motivated by reflection principles. Intuitively, such a cardinal represents a point beyond which initial pieces of the universe of sets start to look similar, in the sense that each is elementarily embeddable into a later one.

Definition

For every ordinal η, a cardinal κ is called η-extendible if for some ordinal λ there is a nontrivial elementary embedding j of Vκ+η into Vλ, where κ is the critical point of j, and as usual Vα 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).

Properties

For a cardinal

\kappa

, say that a logic

L

is

\kappa

-compact if for every set

A

of

L

-sentences, if every subset of

A

or cardinality

<\kappa

has a model, then

A

has a model. (The usual compactness theorem shows

\aleph0

-compactness of first-order logic.) Let
2
L
\kappa
be the infinitary logic for second-order set theory, permitting infinitary conjunctions and disjunctions of length

<\kappa

.

\kappa

is extendible iff
2
L
\kappa
is

\kappa

-compact.[1]

Variants and relation to other cardinals

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 Vκ+η to some Vλ with critical point κ) such that furthermore, Vj(κ) is Σn-correct in V. That is, for every Σn formula φ, φ holds in Vj(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).

See also

References

. Akihiro Kanamori. 2003. Springer. The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings. The Higher Infinite . 2nd. 3-540-00384-3.

Notes and References

  1. Magidor . M. . Menachem Magidor. On the Role of Supercompact and Extendible Cardinals in Logic. 1971. 147—157. Israel Journal of Mathematics. 10. 2. 10.1007/BF02771565 . free.