The famous theorem of
MacDowell and Specker asserts that every model of Peano arithmetic has a proper
elementary end-extension. A consequence of their theorem (and its proof) is that
every model of Peano arithmetic of cardinality less than κ has a κ-like elementary
end-extension, and, in addition, if κ is regular, then there is such a κ-like model in
which all classes are definable. However, under the assumption of the existence of a
κ-Kurepa tree, each model of Peano arithmetic of cardinality less than κ does have
a κ-like elementary end-extension in which there are more than κ generic
classes.
