proper forcing

proper forcing
Name	Proper forcing
Field	Set theory
Introduced	1980s
Introduced by	Shelah
Notable	Martin's Axiom, Souslin's hypothesis, P-ideal dichotomy

proper forcing Proper forcing is a class of forcing notions introduced to control countable structures and preserve desirable combinatorial features of models of Erdős-related combinatorics and Gödel-related constructibility analyses. It was developed to allow extensions in which stationary subsets of ω1 behave predictably while coordinating with large-cardinal hypotheses such as Martin's Axiom and compatibility with iterated constructions by Shelah. Proper forcing mediates interactions among classic objects like Aronszajn trees, Souslin trees, and ideals such as those appearing in the P-ideal dichotomy.

Definition

A forcing notion P is defined to be proper when, for every countable elementary submodel M of a sufficiently large structure like H(θ) containing P and every condition p in P ∩ M, there exists an extension q ≤ p that is (M,P)-generic, meaning q forces the generic filter meets every dense subset D of P that belongs to M. The criterion is formulated using countable elementary submodels originating in work on Cohen forcing and refined in contexts involving Kunen and Jech. Properness ensures preservation of stationarity of sets of ordinals such as subsets of ω1 and is central to arguments involving the interaction of forcing with combinatorial principles like Martin's Axiom and consequences studied by Todorcevic.

Examples and basic properties

Standard examples include countably closed forcings, which are proper by virtue of closure properties previously considered by Cohen and Solovay; forcings that add a Cohen real are typically not proper, while others that add a Hechler real or specialize certain trees can be proper under appropriate definitions used by Shelah and Jech. The class of proper forcings is closed under countable support iterations of length less than certain cardinals, a property exploited in constructions by Foreman and Magidor. Proper forcing notions often preserve cardinals and cofinalities relevant to objects studied by Easton and interact with reflection principles investigated by Todorcevic and Harrison.

Basic properties: every proper forcing preserves ω1 provided there is no collapse forced by conditions in P, a fact used in analyses involving the Continuum Hypothesis and models produced by iterated proper forcing by Shelah and Kunen. Properness can be witnessed by the existence of club many countable elementary submodels satisfying a diagonalization property, a technique found in arguments by Jech and Abraham. Many commonly used forcing notions, such as tree forcings to add or destroy Souslin trees, are either proper or can be made proper by side conditions inspired by Todorcevic.

Preservation and iteration

A central theorem asserts that countable support iterations of proper forcing notions yield a proper forcing notion, under hypotheses regarding uniform definability and parameters as elaborated by Shelah in his iteration theory. Iteration preserves stationarity of subsets of ω1 and preserves chains related to ideals encountered in the study by Farah and Neeman. The proof techniques rely on bookkeeping with countable elementary submodels and diagonalization methods familiar from work by Jech and Kunen. Finite support iterations need not preserve properness; this distinction underpins separation results such as those by Shelah and Jensen.

Preservation results extend to more subtle properties: proper forcings that are ω1-proper or that have the countable covering property with respect to models like L[x] can be iterated with revised countable support to control cardinal arithmetic studied by Easton and Silver. Side condition techniques developed by Mitchell and Neeman refine iteration schemes by including models or structures as part of conditions, enabling longer iterations while retaining properness and preserving combinatorial properties relevant to Martin's Maximum.

Applications in set theory

Proper forcing is instrumental in constructing models satisfying consistency results about the size and behaviour of the continuum considered by Cohen, resolution of statements like the nonexistence of Souslin lines under additional axioms studied by Kunen, and the establishment of consistency of combinatorial dichotomies such as the P-ideal dichotomy and consequences for compact spaces examined by Todorcevic. It underlies proofs of consistency results about reflection principles investigated by Velickovic and produces models with controlled behaviour of stationary reflection as analyzed by Cummings and Magidor.

Proper forcing also facilitates fine structural analysis of definable sets and ideals used in descriptive set theory per Kechris and interactions with determinacy principles studied by Martin. Advanced applications include constructions of models with strong forcing axioms like Martin's Maximum and consequences for the structure of Boolean algebras examined by Todorcevic and Shelah.

Variants and generalizations

Several refinements and generalizations exist: semiproper forcing introduced by Shelah weakens requirements to handle models of size ω1; proper forcing with side conditions due to Mitchell and Neeman incorporates trees of models or models as side conditions; and notions like ω-properness, stationary set preserving forcing, and proper forcing over larger cardinals adapt the concept to different combinatorial contexts studied by Foreman, Magidor, and Väänänen. Further extensions connect to iteration technologies like revised countable support and forcings that are proper relative to a parameter or a class as developed in the work of Shelah and Jech.

Category:Set theory