
TL;DR
This paper proves new theorems about ladder mice and their definability over certain models of set theory, providing a novel proof that avoids stationary towers and extending results to various gap scenarios.
Contribution
It introduces a new analysis of ladder mice and their generalizations, offering a novel proof of the mouse set theorem and extending anti-correctness results to broader contexts.
Findings
Existence of mice with specific definability properties over $J_eta( eals)$
A new proof of the mouse set theorem avoiding stationary towers
Anti-correctness results on a cone at various gap endpoints
Abstract
Assume ZF + AD + . We prove some "mouse set" theorems, for definability over where is a projective-like gap (of ) and is either a successor ordinal or has countable cofinality, but where ends a strong gap. For such ordinals and integers , we show that there is a mouse with . The proof involves an analysis of ladder mice and their generalizations to . This analysis is related to earlier work of Rudominer, Woodin and Steel on ladder mice. However, it also yields a new proof of the mouse set theorem even at the least point where ladder mice arise -- one which avoids the stationary tower. The analysis also yields a corresponding "anti-correctness" result on a cone, generalizing facts familiar in…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Computability, Logic, AI Algorithms · Advanced Graph Theory Research
