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The formal verification of the ctm approach to forcing

by   Emmanuel Gunther, et al.

We discuss some highlights of our computer-verified proof of the construction, given a countable transitive set-model M of 𝑍𝐹𝐢, of generic extensions satisfying 𝑍𝐹𝐢+𝐢𝐻 and 𝑍𝐹𝐢+𝐢𝐻. Moreover, let β„› be the set of instances of the Axiom of Replacement. We isolated a 21-element subset Ξ©βŠ†β„› and defined β„±:β„›β†’β„› such that for every Ξ¦βŠ†β„› and M-generic G, M𝑍𝐢βˆͺβ„±β€œΞ¦βˆͺΞ© implies M[G]𝑍𝐢βˆͺΞ¦βˆͺ{𝐢𝐻}, where 𝑍𝐢 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.


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