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Small measurable cardinals.

Yair Hayut1, Asaf Karagila2

  • 1Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 9190401 Israel.

Archive for Mathematical Logic
|April 20, 2026
PubMed
Summary
This summary is machine-generated.

Researchers improved the definition of j-decomposable systems to better lift elementary embeddings. This work shows the first measurable cardinal can be the first weakly critical or Mahlo cardinal under specific conditions.

Keywords:
Axiom of ChoiceElementary embeddingsMeasurable cardinalsSilver criterionSymmetric extensions

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Area of Science:

  • Set Theory
  • Large Cardinal Theory

Background:

  • The study builds upon prior work on j-decomposable systems and elementary embeddings.
  • Understanding the relationships between different types of large cardinals is a key area in set theory.

Purpose of the Study:

  • To refine the definition of j-decomposable systems for improved lifting properties.
  • To investigate the relationships between measurable, weakly critical, and Mahlo cardinals.
  • To explore the Mitchell order of inaccessible cardinals in specific inner models.

Main Methods:

  • A refined definition of j-decomposable systems is introduced.
  • The improved definition facilitates the lifting of elementary embeddings to symmetric extensions.
  • This lifting criterion is used to analyze the properties of large cardinals.

Main Results:

  • The improved lifting criterion allows for easier handling of weakly compact embeddings.
  • It is demonstrated that the first measurable cardinal can coincide with the first weakly critical cardinal or the first Mahlo cardinal.
  • These findings are relative to the existence of a single measurable cardinal.

Conclusions:

  • The study provides a more effective method for lifting embeddings, advancing large cardinal theory.
  • New relationships between different large cardinal properties are established.
  • The research offers insights into the structure of inner models concerning inaccessible and measurable cardinals.