## 580 -Cardinal arithmetic (11)

March 12, 2009

4. Strongly compact cardinals and ${{\sf SCH}}$

Definition 1 A cardinal ${\kappa}$ is strongly compact iff it is uncountable, and any ${\kappa}$-complete filter (over any set ${I}$) can be extended to a ${\kappa}$-complete ultrafilter over ${I.}$

The notion of strong compactness has its origin in infinitary logic, and was formulated by Tarski as a natural generalization of the compactness of first order logic. Many distinct characterizations have been found.