Set theory: Large Cardinals

Syllabus

Large cardinal axioms postulate the existence of cardinals with a given degree of transcendence over smaller cardinals and provide a superstructure for the analysis of strong mathematical statements. The investigation of these axioms is indeed a mainstream of modern set theory. For instance, they play a crucial role in the study of definable sets of reals and their regularity properties such as Lebesgue measurability. Although formulated at various stages in the development of set theory and with different motivation, these hypotheses were found to form a linear hierarchy reaching up to the inconsistency. All known set-theoretic propositions can be gauged in this hierarchy in terms of their consistency strength, and the emerging structure of implications provides a remarkably rich, detailed and coherent picture of the strongest propositions of mathematics as embedded in set theory.

Contents

• Inaccessible, compact, measurable cardinals
• Partition properties and trees
• The indescernibles and 0#
• Sets of reals, Lebesgue measurability, property of Baire
• Iterated elementary embeddings
• Very large cardinals: supercompact, huge...
• Determinacy of games

Bibliography

- P. Dehornoy: Théorie des ensembles: Introduction à une théorie de l'infini et des grands cardinaux (Calvage Mounet 2017)
- T. Jech: Set theory, The Third Millennium Edition (Springer Monographs in Mathematics 2003)
- A. Kanamori : The higher infinite: Large cardinals in set theory from their beginnings (Springer Verlag, 2003)

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