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Transcript of Kunen.7.E4
Fix a model M of GCH. Let I be uncountable in M , let P = I,2, and let G be P-generic over M . LetN = L(P(ω))M [G], the constructible hull in M [H] around P(ω) ∩M [H]. We argue N does not modelchoice, though since M [G] models ZFC, N must model ZF; this would imply Con(ZF)→ Con(ZF+¬AC).
So, suppose N models AC, and let κ be an ordinal in N (necessarily, κ ∈ M as well, since forcingadds no ordinals) such that (κ = |P(ω)|)L(P(ω)) in M [G]. Now take λ ∈ M which is a regular cardinalcontaining κ in M , let Q = Fn(λ, 2), and let H be Q-generic over M .
By E1, E2, and E3, we know that for any ordinal α (including κ) in M , and any formula φ(x),φ(α)L(P(ω)) is true in M [G] if and only if 1 P φ(α̌) if and only if 1 Q φ(α̌) if and only if φ(α)L(P(ω)) istrue in M [H]; thus, in M [H], (κ = |P(ω)|)L(P(ω)).
But κ ∈ λ (absolutely, in every transitive model) so, by preservation of cardinals, κ is not in bijectionwith λ in M [H]. But λ = |P(ω)| in M [H], since GCH holds in M , λ is regular in M , and we are forcingover Fn(λ × ω, 2), so there is a bijection there. But by preservation of (κ = |P(ω)|)L(P(ω)), there is abijection (constructible over P(ω)) between κ and P(ω), violating the cardinal-hood of λ in M [H], acontradiction.
It should be noted that N is a model of V = L(P(ω)), by absoluteness of P and D. It is thus alsoa model of V = HOD(P(ω)), since ZFC proves ∀X(L(X) ⊂ HOD(X)). So, our actual implication isCon(ZF)→ Con(ZF + ¬AC + V = L(P(ω)) + V = HOD(P(ω))).