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Published online by Cambridge University Press: 12 March 2014
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 κκ/G is not well-founded.
Theorem. Let G be Spector-generic over a model M of 
, for all α < κ.
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 Ae M. Let Φ = Κκ/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 
for all α <ℵ1 and all ordinals β.
5) There is a linear order Φ′ with property 3) above which is not “weakly compact”, i.e., Φ′ ↛ (Φ′)2.
