Unforcing and the Ground Axiom
My interest in “unforcing,” somehow working backwards from a forcing extension to a ground model, started early in my set theory studies. When I went through the rite of passage of learning Paul Cohen’s forcing – an exercise in induction, with a lovely (and at the time, frustratingly opaque) back-and-forth between technical details and philosophical issues – my initial reaction was “This is a great operation. What’s the inverse?” The resulting discussions with my advisor, Joel David Hamkins, led us eventually to the formulation of the Ground Axiom, a first attempt to explore seriously an aspect of unforcing.
In my dissertation I focussed on models in which the universe is not a forcing extension of an inner model (we say such a model satisfies the Ground Axiom): In order to understand how to “unforce,” let’s first see what it’s like to not be able to do so. I also briefly considered
Class forcing from scratch
One of the main tools I used in this work was class forcing, both iterations and products. As a new student of forcing, I was unsatisfied with the expositions of class forcing available in the standard texts – many of them seemed to rely on the offhand “this proof is readily adapted to the class context” or “the class version of this lemma is immediate.” With that in mind, I included an appendix that develops and proves the main results of class forcing, mirroring the structure for development of set forcing but presenting the main arguments in their entirety. Unlike the case of set forcing, for which every partial order preserves ZFC, class forcing requires some additional restrictions to ensure ZFC in the extension, and I focussed on two such restrictions (progressively closed products and iterations) that cover a large variety of class forcings including all those employed in the dissertation.
Details
Abstract. A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set-forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class-forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set-forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent. As many of these results rely on forcing with proper classes, an appendix is provided giving an exposition of the underlying theory of proper class forcing.
Reitz, Jonas, “The Ground Axiom” (2006). CUNY Academic Works, arXiv.
Featured image by Oleg Mitiukhin on Unsplash
