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Global Chang's Conjecture and singular cardinals.

Monroe Eskew1, Yair Hayut2

  • 1Institut für Mathematik, Kurt Gödel Research Center, Universität Wien, Kolingasse 14-16, 1090 Wien, Austria.

European Journal of Mathematics
|November 1, 2021
PubMed
Summary

This study explores global versions of Chang's Conjecture involving singular cardinals. Results show limitations and demonstrate consistency for Chang's Conjecture between limit cardinals below a certain value, relative to large cardinals.

Keywords:
Chang’s ConjectureDiagonal Prikry forcingScales

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Area of Science:

  • Set Theory
  • Large Cardinal Theory

Background:

  • Chang's Conjecture is a significant statement in set theory.
  • Investigating its global versions and implications for singular cardinals is crucial for understanding set-theoretic universe.

Purpose of the Study:

  • To explore global versions of Chang's Conjecture involving singular cardinals.
  • To establish limitations on these principles.
  • To prove the consistency of Chang's Conjecture between limit cardinals relative to large cardinals.

Main Methods:

  • Utilizing techniques from large cardinal theory.
  • Employing model-theoretic arguments to establish consistency results.

Main Results:

  • Demonstration of limitations on global versions of Chang's Conjecture.
  • Proof of the relative consistency of Chang's Conjecture holding between all pairs of limit cardinals below , assuming the existence of large cardinals.

Conclusions:

  • The findings provide new insights into the behavior of Chang's Conjecture in the presence of singular cardinals.
  • The results contribute to the understanding of the consistency strength of set-theoretic principles.