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Determinacy Axioms and Large Cardinals

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Logic and Its Applications (ICLA 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13963))

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Abstract

The study of inner models was initiated by Gödel’s analysis of the constructible universe. Later, the study of canonical inner models with large cardinals, e.g., measurable cardinals, strong cardinals or Woodin cardinals, was pioneered by Jensen, Mitchell, Steel, and others. Around the same time, the study of infinite two-player games was driven forward by Martin’s proof of analytic determinacy from a measurable cardinal, Borel determinacy from ZFC, and Martin and Steel’s proof of levels of projective determinacy from Woodin cardinals with a measurable cardinal on top. First Woodin and later Neeman improved the result in the projective hierarchy by showing that in fact the existence of a countable iterable model, a mouse, with Woodin cardinals and a top measure suffices to prove determinacy in the projective hierarchy. This opened up the possibility for an optimal result stating the equivalence between local determinacy hypotheses and the existence of mice in the projective hierarchy. This article outlines the main concepts and results connecting determinacy hypotheses with the existence of mice with large cardinals as well as recent progress in the area.

Supported by L’ORÉAL Austria, in collaboration with the Austrian UNESCO Commission and in cooperation with the Austrian Academy of Sciences - Fellowship Determinacy and Large Cardinals and the Austrian Science Fund (FWF) under Elise Richter grant number V844, international project number I6087, and START grant number Y1498.

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  1. Institut für Diskrete Mathematik und Geometrie, TU Wien, Wiedner Hauptstraße 8-10/104, 1040, Wien, Austria

    Sandra Müller

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  1. Indian Institute of Technology Kanpur, Kanpur, India

    Mohua Banerjee

  2. Indian Institute of Technology Goa, Ponda, India

    A. V. Sreejith

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Müller, S. (2023). Determinacy Axioms and Large Cardinals. In: Banerjee, M., Sreejith, A.V. (eds) Logic and Its Applications. ICLA 2023. Lecture Notes in Computer Science, vol 13963. Springer, Cham. https://doi.org/10.1007/978-3-031-26689-8_5

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