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Recursion Theory by Liang Yu, Chi Tat Chong

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6.2.5 Destroying the Axiom of Choice

In general there is no direct way of destroying the Axiom of Choice (AC) by a forcing extension of a transitive model of ZFC. One introduces an intermediate model to achieve this.

Definition 6.2.13.

–A set X is ordinal definable in M, written X ∈ ODM, if there is a formula φ(x, v0,…, vn) together with ordinals α0,…, αn, β in M such that aXimageφ(a, α0,…, αn), where image is the collection of sets in M of rank less than β.

X is hereditarily ordinal definable in M, written X ∈ HODM, if TC({X}) ⊂ ODM.

If M is a transitive ...

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