Abstract

Forcing with [κ] κ over a model of set theory with a strong partition cardinal, M. Spector produced a generic ultrafilter G on κ such that κ κ /G is not well-founded. Theorem. Let G be Spector-generic over a model M of $ZF + DC + \kappa \rightarrow (\kappa)^\kappa_\alpha, \kappa > \omega$ , for all $\alpha . 1) Every cardinal (well-ordered or not) of M is a cardinal of M[ G]. 2) If A ∈ M[ G] is a well-ordered subset of M, then A ∈ M. Let Φ = κ κ /G. 3) There is an ultrafilter U on Φ such that every member of U has a subset of type Φ, and the intersection of any well-ordered subset of U is in U. 4) Φ satisfies Φ → (Φ) α β for all $\alpha and all ordinals β. 5) There is a linear order Φ' with property 3) above which is not "weakly compact", i.e., $\Phi' \nrightarrow (\Phi')^2$