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Preprint Number 801
801. Itay Kaplan, Saharon Shelah
Forcing a countable structure to belong to the ground model
Submission date: 25 November 2014.
Suppose that P is a forcing notion, L is a language (in V), τ' a P-name such that P ⊩ τ' is a countable L-structure. In the product P × P, there are names τ_1',τ_2' such that for any generic filter G = G_1 × G_2 over P × P, τ'_1[G]=τ'[G_1] and τ'_2[G]=τ'[G_2]. Zapletal asked whether or not P × P ⊩ τ'_1 ≅ τ'_2 implies that there is some M in V such that P ⊩ τ' ≅ Mˇ. We answer this negatively and discuss related issues.
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