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Published online by Cambridge University Press: 18 December 2013
Smooth equivalence relations
In the case that the equivalence relation E is smooth, the model V[xgen]E as described in Theorem 3.5 takes on a particularly simple form.
Proposition 4.1Let I be a σ-ideal on a Polish space X such that the quotient forcing PI is proper, and suppose that E is a smooth equivalence relation on the space X. Then V[xgen]E = V[f(ẋgen)] for every ground model Borel function f reducing E to the identity.
Proof Choose a Borel function f: X → 2ω that reduces the equivalence E to the identity. Let G ⊂ PI and H ⊂ Coll(ω, k) be mutually generic filters, and let x ∈ X be a point associated with the filter G. First of all, f (x) ∈ 2ω is definable from the equivalence class [x]E in the model V[x][H]: it is the unique value of f(x) for all x ∈[xgen]E. Thus, V[f (x)]⊂ V[x]E. On the other hand, the equivalence class [x]E is also definable from f(xgen), the model V[G, H] is an extension of V[f(x)] via a weakly homogeneous notion of forcing Coll(ω, k), and therefore by Fact 2.24, V[xgen]E ⊂ V[f(x)].
The total canonization of smooth equivalence relations has an equivalent restatement with the quotient forcing adding a minimal real degree.
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