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Ebook: Set Theory: Exploring Independence and Truth

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27.01.2024
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This textbook gives an introduction to axiomatic set theory and examines the prominent questions that are relevant in current research in a manner that is accessible to students. Its main theme is the interplay of large cardinals, inner models, forcing and descriptive set theory.

The following topics are covered:

• Forcing and constructability
• The Solovay-Shelah Theorem i.e. the equiconsistency of ‘every set of reals is Lebesgue measurable’ with one inaccessible cardinal
• Fine structure theory and a modern approach to sharps
• Jensen’s Covering Lemma
• The equivalence of analytic determinacy with sharps
• The theory of extenders and iteration trees
• A proof of projective determinacy from Woodin cardinals.

Set Theory requires only a basic knowledge of mathematical logic and will be suitable for advanced students and researchers.




Set theory aims at proving interesting true statements about the mathematical universe. Different people interpret ‘‘interesting’’ in different ways. It is well known that set theory comes from real analysis. This led to descriptive set theory, the study of properties of definable sets of reals, and it certainly is an important area of set theory. We now know that the theory of large cardinals is a twin of descriptive set theory. I find the interplay of large cardinals, inner models, and properties of definable sets of reals very interesting. We give a complete account of the Solovay-Shelah Theorem according to which having all sets of reals to be Lebesgue measurable and having an inaccessible cardinal are equiconsistent. We give a modern account of the theory of 0#, produce Jensen’s Covering Lemma, and prove the Martin-Harrington Theorem according to which the existence of 0# is equivalent with E11 determinacy. We also produce the Martin-Steel Theorem according to which Projective Determinacy follows from the existence of infinitely many Woodin cardinals.


Set theory aims at proving interesting true statements about the mathematical universe. Different people interpret ‘‘interesting’’ in different ways. It is well known that set theory comes from real analysis. This led to descriptive set theory, the study of properties of definable sets of reals, and it certainly is an important area of set theory. We now know that the theory of large cardinals is a twin of descriptive set theory. I find the interplay of large cardinals, inner models, and properties of definable sets of reals very interesting. We give a complete account of the Solovay-Shelah Theorem according to which having all sets of reals to be Lebesgue measurable and having an inaccessible cardinal are equiconsistent. We give a modern account of the theory of 0#, produce Jensen’s Covering Lemma, and prove the Martin-Harrington Theorem according to which the existence of 0# is equivalent with E11 determinacy. We also produce the Martin-Steel Theorem according to which Projective Determinacy follows from the existence of infinitely many Woodin cardinals.

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