Appendix

Summary

This appendix is independent of the rest of the book and contains properties of a model obtained by adding a class generic G which adds a Cohen generic to every successor cardinal. The results of the appendix are used in §4.2 which contains the definitions and elementary properties of the large cardinal properties used here. We shall prove Facts 4.2.13–15. Though much of the material presented here (if not all) is known it has never appeared in print.

We start with a model M |= ZF + GCH. Let Q be the set of conditions s.t.

∈ Q iff p: dom(p) → 2 where dom(p) ∩ [ω,∞) and card(dom(p/K)) < K for all regular cardinals K.

We shall use the following notation:

Hence Q^α are the standard Cohen conditions for adding a subset to [α, α⁺) card = {α| α is an infinite cardinal}.

For α ∈ card, Q_αis α-closed. card(Q^α) = α so in general Q^α is ⁺–AC. But for Mahlo cardinals, K, Q^α is K−AC.

Lemma A.I Let K be Mahlo, then Q^K; is K−AC.

Proof Let Δ ⊆ Q^K; be a set of mutually incompatible conditions. By Mahloness there is a regular s.t. is a set of mutually incompatible conditions. We claim then p ∈ Δ/Δ must be mutually incompatible with all of contradicting the maximality of.

Set N = MCA], where A is Q-generic over M. Set A^α = A ∩ α and A^α = A ∩ [α,∞). Clearly N |= ZF + GCH in which cardinals and cofinalities are preserved and universal choice holds in N (Easton [E]).