Strong cardinal

In set theory, a strong cardinal is a type of large cardinal. It is a weakening of the notion of a supercompact cardinal.

Formal definition

If λ is any ordinal, κ is λ-strong means that κ is a cardinal number and there exists an elementary embedding j from the universe V into a transitive inner model M with critical point κ and

${\displaystyle V_{\lambda }\subseteq M}$

That is, M agrees with V on an initial segment. Then κ is strong means that it is λ-strong for all ordinals λ.

Relationship with other large cardinals

It is obvious from the definitions that strong cardinals lie below supercompact cardinals and above measurable cardinals in the consistency strength hierarchy.

They also lie below superstrong cardinals and Woodin cardinals. However, the least strong cardinal is larger than the least superstrong cardinal.

References

• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3.