A -extendible cardinal is a type of large cardinal which generalizes extendible cardinals. They come up in formulating refined versions of Vopěnka's principle.
We work in ZFC. Let denote the class of infinite cardinal numbers such that (see von Neumann hierarchy) is a -elementary submodel of the universe , i.e. such that the embedding preserves and reflects the truth of -formulas. In particular:
is if and only if it is infinite, and
is if and only if it is uncountable and coincides with the hereditarily -sized sets , i.e. those whose transitive closure has cardinality .
If and both are in , we say is --extendible if there is an elementary embedding with critical point , such that , , and .
We say is -extendible if it is --extendible for all with .
- is -extendible if and only if it is extendible.
Created on September 24, 2012 06:15:34
by Mike Shulman