Levy collapse

Levy collapse The Levy collapse is a family of set-theoretic forcing notions originally defined to collapse cardinals to become equal to smaller cardinals; it is instrumental in constructions involving Paul Cohen-style forcing, Kurt Gödel-style inner models, and large cardinal hypotheses such as Measurable cardinals and Supercompact cardinals. The technique interacts with combinatorial principles like Diamond principle, structural frameworks like L, and global constraints exemplified by Easton's theorem and Martin's Axiom, and is frequently used alongside forcing notions such as Cohen forcing, Add(κ,1), and Levy collapses in independence proofs.

Definition and basic construction

A Levy collapse is typically denoted Coll(κ, <λ) or Coll(κ, λ) and is defined in the context of a ground model such as ZFC or extensions with AC; the poset consists of partial functions with bounded domain and finite or bounded conditions that collapse a cardinal λ down to have cofinality or cardinality κ. The standard presentation uses conditions that are functions from ordinals below λ to κ with finite support, following frameworks present in the literature of Azriel Levy and later expositions by researchers connected to Dana Scott, Robert Solovay, and Kenneth Kunen. In typical use one forces with Coll(ω, <κ) to make κ become ℵ1 or with Coll(κ, <λ) to set λ to κ+, aligning with constructions in papers by William Mitchell and lectures by Jech.

Variants and common forcings

Common variants include the collapse Coll(ω, κ), the bounded-support collapse Coll(κ, <λ), and Easton-style product collapses that combine collapses at many cardinals; these are often compared to Cohen forcing, Random forcing, and Prikry forcing in their combinatorial and preservation behavior. Iterated forms, such as Easton-support iterations and finite-support products, appear in works by Easton, Solovay, and Silver, and are used to produce models like those constructed by Magidor for changing cofinalities while preserving large cardinals such as Supercompact cardinals or Measurable cardinals. Specialized collapses adapted to cardinals with combinatorial features, for example collapses preserving Mitchell order structure or singular cardinals in the style of Gitik, are standard tools in contemporary research.

Preservation and failure of set-theoretic properties

Levy collapses have nuanced preservation properties: they often preserve cardinals below the collapse point while collapsing targeted cardinals, but can destroy combinatorial principles like the Square principle or Chang's conjecture unless carefully controlled. Preservation theorems relate to absoluteness results for L and interactions with large cardinal indestructibility results such as those of Laver and Hamkins. Failure phenomena include the destruction of measurability for a Measurable cardinal under small collapses, singular cardinal combinatorics changes studied by Shelah, and alterations to the structure of Ultrafilters and normal measures investigated by Kunen and Mitchell.

Applications in independence proofs

Levy collapses are central to classical independence proofs, including early proofs that certain combinatorial statements are independent of ZFC when combined with forcing by Cohen or Random reals, and later proofs establishing relative consistency results about failure or satisfaction of instances of GCH or failures of the Singular Cardinals Hypothesis by researchers like Easton, Silver, and Gitik. They play key roles in establishing relative consistency of statements about the existence or nonexistence of definable well-orders in HOD and in constructions that separate large cardinal axioms from combinatorial consequences, as in work by Magidor, Woodin, and Foreman. Levy collapses also appear in consistency proofs concerning forcing axioms such as Martin's Maximum and their consequences for the structure of the continuum.

Historical development and key results

The technique traces to Azriel Levy's investigations in the 1960s, with foundational developments by Cohen's forcing revolution, followed by systematic analyses by Solovay, Easton, and Silver in the study of cardinal arithmetic and independence of GCH. Subsequent milestones include Laver's indestructibility results, Magidor's changing cofinalities via collapses, Gitik's singular cardinal constructions, and Woodin's applications in determinacy and inner model theory; each of these built on collapsing techniques refined by Kunen, Mitchell, and Shelah. Contemporary research connects Levy-style collapses to inner model programs involving iterable mice and to applications in descriptive set theory by researchers affiliated with institutions like Princeton University, University of Oxford, and Hebrew University of Jerusalem.

Category:Set theory