Small measurable cardinals

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.