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4 Fix a model M of GCH. Let I be uncountable in M , let P = I,2, and let G be P-generic over M . Let N = L(P (ω)) M[G] , the constructible hull in M [H] around P (ω) M [H]. We argue N does not model choice, 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 forcing adds no ordinals) such that (κ = |P (ω)|) L(P(ω)) in M [G]. Now take λ M which is a regular cardinal containing 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 φα) if and only if 1 Q φα) if and only if φ(α) L(P(ω)) is true 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 bijection with λ in M [H]. But λ = |P (ω)| in M [H], since GCH holds in M , λ is regular in M , and we are forcing over Fn(λ × ω, 2), so there is a bijection there. But by preservation of (κ = |P (ω)|) L(P(ω)) , there is a bijection (constructible over P (ω)) between κ and P (ω), violating the cardinal-hood of λ in M [H], a contradiction. It should be noted that N is a model of V = L(P (ω)), by absoluteness of P and D. It is thus also a model of V = HOD(P (ω)), since ZFC proves X(L(X) HOD(X)). So, our actual implication is Con(ZF) Con(ZF + ¬AC + V = L(P (ω)) + V = HOD(P (ω))). 1

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Solution to Kunen - Set Theory, 1st Edition, chapter 7, exercise E4

### Transcript of Kunen.7.E4 4

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))).

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