iterable mice
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| iterable mice | |
|---|---|
| Name | Iterable mice |
| Type | Inner model objects |
| Field | Set theory |
| Introduced | 1970s–1990s |
| Notable | Donald A. Martin, John R. Steel, W. Hugh Woodin, Akihiro Kanamori, William J. Mitchell, Rudolf M. Ketchersid |
iterable mice Iterable mice are fine-structured transitive models of fragments of Zermelo–Fraenkel set theory enriched with sequences of extenders and measures that permit coherent comparison via iteration trees. They serve as canonical approximations to large cardinal strength inside core models and mediate interactions among descriptive set theory, determinacy, and large cardinals. Constructions of iterable mice use techniques from ultrapower embeddings, fine structure theory, and comparison processes that trace back to foundational work by researchers associated with inner model theory and determinacy programs.
Definition and Basic Properties
An iterable mouse is typically described as a structure (M, ∈, E, ...) satisfying fine structural axioms and possessing iteration strategies that yield comparison success. Foundational contributors include Donald A. Martin, John R. Steel, W. Hugh Woodin, Akihiro Kanamori, William J. Mitchell and Rudolf M. Ketchersid. Basic properties often referenced in the literature are solidity, soundness, condensation, and coherence of extender sequences, all studied in settings influenced by Kurt Gödel's constructible hierarchy and extensions such as Solovay models. Iterability is formulated via trees and strategies related to notions developed by Mitchell and Steel in comparison lemmas and uniqueness results.
Construction and Fine Structure
Constructions proceed by iterating extenders to build levels analogously to the Gödel constructible universe L but using extender sequences. Key techniques derive from work by Rudolf M. Ketchersid, Mitchell, Steel, and Hugh Woodin on extender models and the development of Jensen-style fine structure adapted to extenders. Canonical inner models like the Dodd–Jensen core and Mitchell–Steel models are landmarks; related constructions reference the Dodd–Jensen lemma, Mitchell order, and the theory of coherent sequences of measures. Technical apparatus includes comparison of ultrapowers introduced by Solovay, extenders modeled on elementary embeddings studied by Ken Kunen and Akihiro Kanamori, and strategies whose analysis uses ideas from the Axiom of Determinacy investigations by Martin and Steel.
Iterability and Comparison Processes
Iterability is certified by the existence of iteration strategies that guide successful runs of iteration trees and produce wellfounded models; this framework was elaborated in the comparison theorem of Mitchell and Steel. Comparison processes rely on fine structural facts and factor embeddings reminiscent of the techniques used by Donald A. Martin in determinacy proofs and by Hugh Woodin in analysis of strong cardinals. Iteration trees and stacks appear in works by John R. Steel, William J. Mitchell, and Ronald Jensen and are linked to proof techniques from combinatorial set theory explored by Akihiro Kanamori and Ken Kunen. Critical notions include branch condensation, branch uniqueness, and the Dodd–Jensen core model comparison developed by Jensen and later extended by Steel.
Types and Examples of Mice
Important explicit classes include 0#, small mice with finitely many Woodin cardinals, Mitchell–Steel mice, and hybrid mice incorporating predicates from forcing extensions. Prototype examples arise in the analysis of the minimal models accommodating measurable cardinals (linked historically to Solovay and Ulam ideas) and models with Woodin cardinals studied by Hugh Woodin and John R. Steel. Specific named constructions frequently cited include the Mitchell–Steel core models, the Dodd–Jensen core, and canonical mice used in determinacy proofs by Donald A. Martin and John R. Steel. Additional exemplar models are those appearing in the work of Rudolf M. Ketchersid, Ronald Jensen, Akihiro Kanamori, William J. Mitchell, and later refinements by Brent Hechler-style constructions adapted to inner model contexts.
Role in Inner Model Theory and Determinacy
Iterable mice are central to inner model theory’s program to internalize large cardinal hypotheses and to connect these with descriptive set theory results such as projective determinacy. Seminal interactions involve the Axiom of Determinacy research of Donald A. Martin and the core model induction methods developed by John R. Steel and Hugh Woodin. The analysis of scales and pointclasses in descriptive set theory uses canonical mice to witness determinacy for classes studied by Martin and Steel, while large cardinal exactness is pursued in collaboration with work by Akihiro Kanamori and Hugh Woodin. Results tying mice to consequences like determinacy, regularity properties, and scales reference comparisons and iterability criteria established by Mitchell, Jensen, and Steel.
Related Techniques and Applications
Techniques related to iterable mice include tree forcing analogues, comparison lemmas, and core model induction. Applications touch on the proof of projective determinacy linked to strategies by Donald A. Martin and John R. Steel, analyses of inner models with Woodin cardinals due to Hugh Woodin, and consistency lower bounds studied by Akihiro Kanamori and Rudolf M. Ketchersid. Other adjacent methods involve extenders and ultrapower maps traced to Ken Kunen and Solovay and fine-structural condensation principles developed by Ronald Jensen and William J. Mitchell. Contemporary research integrates these tools with descriptive set theory programs advanced by Martin, Steel, and Woodin.