L(R)

L(R)
Name	L(R)
Subject	Set theory
Created	1970s–1980s
Key figures	Kurt Gödel, Robert M. Solovay, John R. Steel, Donald A. Martin, W. Hugh Woodin
Major results	Determinacy consequences, regularity of sets of reals, inner model relationships

L(R)

L(R) is the inner model built over the class of real numbers that plays a central role in modern descriptive set theory, inner model theory, and the study of determinacy. It is a transitive proper class (or model of ZF or ZFC minus Choice depending on hypotheses) containing all ordinals and all real numbers from the ambient universe, and it organizes sets built from reals using definability and the constructibility hierarchy. L(R) serves as a bridge between classical constructibility results of Kurt Gödel and the determinacy-driven analysis associated with Donald A. Martin, John R. Steel, and W. Hugh Woodin.

Definition and Construction

L(R) is defined by iterating the constructible hierarchy while allowing parameters from the set of real numbers, producing a minimal inner model that contains the reals. The construction parallels Gödel’s L but at each stage permits definability with parameters from the class of reals and from previously constructed sets, yielding levels Lα(R) indexed by ordinals α. The model is the union ⋃α Lα(R), capturing sets definable from reals and ordinals; its formation uses notions developed by Solovay, Jensen, and later formalized in the work of Steel and Woodin. Under determinacy or large cardinal hypotheses associated with Measurable cardinal, Woodin cardinal, and Supercompact cardinal axioms, L(R) satisfies strong regularity and closure properties established in the literature by researchers at institutions such as University of California, Berkeley and Princeton University.

Fundamental Properties

L(R) is transitive, contains all ordinals, and is minimal with respect to containing the real numbers and being definable-closed under operations permitted in the construction. When the ambient universe satisfies the Axiom of Choice via models like ZFC, L(R) often fails Choice internally, exhibiting forms of determinacy instead; results of Solovay and Martin show many regularity properties for sets of reals inside L(R), including Lebesgue measurability and the Baire property for projective sets. The interplay of L(R) with canonical inner models such as those constructed by Jensen (for the constructible universe), with core models studied by Mitchell and Steel, and with determinacy models used by Martin and Woodin yields a landscape where notions like scales, norms, and uniformization are analyzed. L(R) often satisfies determinacy axioms like AD or forms of AD+, and its ordinal structure reflects the existence of large cardinals such as Woodin cardinals; key combinatorial features mirror those in HOD analyses and in comparisons with models like the Core model K.

Determinacy and Large Cardinals

A central theme is the equivalence and implication relationships between determinacy axioms in L(R) and large cardinal axioms in the ambient universe. Results by Martin, Steel, and Woodin demonstrate that sufficient large cardinal strength, for example the existence of countably many Woodin cardinals and a measurable above them, yields AD in L(R). Conversely, determinacy assumptions for projective sets in L(R) imply consistency strength that can be calibrated against measurable and Woodin hypotheses investigated by Solovay and later by Welch. The analysis employs iteration trees, extender models, and comparison arguments developed by Mitchell, Dodd, and Steel; these techniques connect determinacy in L(R) with the existence of inner models carrying suitable extenders and with large cardinal indiscernibility from work by Silver and Kun (referring to Kenneth Kunen).

Sets of Reals in L(R)

Sets of reals in L(R) exhibit strong structural and regularity features: under determinacy hypotheses they have the perfect set property, Lebesgue measurability, and the property of Baire as shown in results of Solovay and Martin. Projective hierarchies and pointclasses in L(R) are analyzed via scales, prewellorderings, and norms developed in studies by Moschovakis, Kechris, and Steel; key objects include PD (Projective Determinacy) consequences, analytic sets, and higher projective levels studied in monographs by Kechris and Moschovakis. The study of Wadge degrees, Suslin representations, and uniformization within L(R) draws on work by Wadge, Suslin, and Davis, linking descriptive set-theoretic structure to inner model fine structure from Jensen and extender algebra techniques introduced by Woodin.

Forcing and Independence Results

Forcing over models containing L(R) and forcing arguments relative to L(R) are central to independence proofs connecting determinacy, Choice, and large cardinals. Seminal forcing results by Solovay demonstrated models where all sets of reals are Lebesgue measurable; subsequent arguments by Woodin and Steel use forcing to separate AD, AC, and statements about HOD and L(R). Forcing notions such as collapse forcings, Prikry forcing, and extender-based forcings studied by Magidor, Gitik, and Foreman are applied to modify the relationship between the ambient universe and L(R), producing independence results about scales, singular cardinal behavior, and the failure or preservation of determinacy in generic extensions. Preservation theorems for determinacy in L(R) rely on fine structural analyses by Steel and iterability results stemming from work by Mitchell and Dodd-Jensen.

Historical Development and Key Results

The conception of inner models built from reals evolved from Gödel’s constructible universe and the descriptive set-theoretic program initiated by Suslin and Luzin. In the 1970s and 1980s, breakthroughs by Solovay, Martin, and Steel established determinacy consequences and relationships with measurable cardinals. The 1990s and 2000s saw refinements by Woodin and collaborators connecting AD in L(R) with Woodin cardinals and developing AD+ and hod mouse frameworks; influential theorems include Martin–Steel determinacy results and Woodin’s analysis of the interaction between large cardinals and determinacy. Contemporary research continues at centers such as Institute for Advanced Study and Université Paris, with active contributions from researchers like John R. Steel, W. Hugh Woodin, Grigor Sargsyan, and Ralf Schindler advancing core model and hod analysis relevant to L(R).

Category:Inner models