## On strong forms of extendibility

1. Abstract The large cardinal program studies the properties of large cardinals; cardinals with large cardinal properties. Many large cardinal properties have been discovered: Inaccessibility; Mahloness, weak compactness, indescribability, partition properties, $$0^\sharp$$, measurability, $$0^\dagger$$, and the hierarchy of huge cardinals. The last collection of properties is particularly interesting. The hierarchy of huge cardinals consists of…