First Steps Towards a Formalization of Forcing

Abstract

We lay the ground for an Isabelle/ZF formalization of Cohen's technique of forcing. We formalize the definition of forcing notions as preorders with top, dense subsets, and generic filters. We formalize a version of the principle of Dependent Choices and using it we prove the Rasiowa-Sikorski lemma on the existence of generic filters.Given a transitive set M , we define its generic extension M [G], the canonical names for elements of M , and finally show that if M satisfies the axiom of pairing, then M [G] also does. We also prove M [G] is transitive.

A formal proof of the independence of the continuum hypothesis

“…A large body of formalized set theory has been completed in Isabelle/ZF, led by Paulson and his collaborators [32,33,35], including the relative consistency of AC with ZF [34]. Building on this, Gunther, Pagano, and Terraf have taken some first steps towards formalizing forcing [15,16], by way of generic extensions of countable transitive models.…”

Mechanization of Separation in Generic Extensions

“…The original insight by Cohen was to define the notion of genericity for a filter G ⊆ P and to prove that whenever G is generic, M [G] will satisfy ZF . Remember that a filter is generic if it intersects all the dense sets in M ; in [12] we formalized the Rasiowa-Sikorski lemma which proves the existence of generic filters for ctms.…”

“…In order to use wfrec the relation should be expressed as a set, so in [12] we originally took the restriction of ed to the whole universe M ; i.e. ed ∩ M × M .…”

“…In pursuing the proof of Separation on generic extensions we extended Paulson's library with: (i) renaming of variables for internalized formulas, which with little effort can be extended to substitutions; (ii) an improvement on definitions by recursion on well-founded relations; (iii) enhancements in the hierarchy of locales; and (iv) a variant of the principle of dependent choices and a version of Rasiowa-Sikorski, which ensures the existence of generic filters for countable and transitive models of ZF ; the last item was already communicated in the first report [12].…”

Formalization of Forcing in Isabelle/ZF

“…We work using the implementation of Zermelo-Fraenkel (ZF ) set theory Isabelle/ZF by Paulson and Grabczewski [17]. In an early paper [3], we set up the first elements of the countable transitive model (ctm) approach, defining forcing notions, names, generic extensions, and showing the existence of generic filters via the Rasiowa-Sikorski lemma (RSL). In our second (unpublished) technical report [4] we advanced by presenting the first accurate formal abstract of the Fundamental Theorems of Forcing, and using them to show that the ZF axioms apart from Replacement and Infinity hold in all generic extensions.…”