The formal verification of the ctm approach to forcing

10/27/2022

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by   Emmanuel Gunther, et al.

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We discuss some highlights of our computer-verified proof of the construction, given a countable transitive set-model M of 𝑍𝐹𝐶, of generic extensions satisfying 𝑍𝐹𝐶+𝐶𝐻 and 𝑍𝐹𝐶+𝐶𝐻. Moreover, let ℛ be the set of instances of the Axiom of Replacement. We isolated a 21-element subset Ω⊆ℛ and defined ℱ:ℛ→ℛ such that for every Φ⊆ℛ and M-generic G, M𝑍𝐶∪ℱ“Φ∪Ω implies M[G]𝑍𝐶∪Φ∪{𝐶𝐻}, where 𝑍𝐶 is Zermelo set theory with Choice. To achieve this, we worked in the proof assistant Isabelle, basing our development on the Isabelle/ZF library by L. Paulson and others.