Abstract
This part contains brief informal discussions (with proofs and most details omitted) of some of the landmark results of set theory of the past 75 years. Topics discussed are constructibility, forcing and independence results, large cardinal axioms, infinite games and determinacy, projective determinacy, and the status of the Continuum Hypothesis.
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Notes
- 1.
Gödel’s method of showing relative consistency, known as the method of inner models, cannot be used to show the relative consistency of the negation of CH (or of the negation of any statement provable from V=L). The reason is that the only inner model of L containing the ordinals is L itself.
- 2.
Similar to the search for discovering true principles in physics.
- 3.
Set theorists differ widely on these matters, and pluralists and believers of Gödel’s program represent only two of many possible viewpoints. Feferman has expressed that the Continuum Hypothesis is not even a definite mathematical problem. See [16] for a panoramic debate, [50] for some background, and [51] for more references. See also the EFI project web site http://logic.harvard.edu/efi.php.
- 4.
An earlier result of Scott, D.—nnnScott had shown that the axiom of constructibility contradicts the existence of measurable cardinals. Gaifman, H.—nnnGaifman, Rowbottom, F.—nnnRowbottom and Silver, J.—nnnSilver dramatically improved Scott’s result to show that if a measurable cardinal exists then in a certain sense the vast majority of sets must be non-constructible.
- 5.
Other stronger structural properties that we have not defined (such as reduction, pre-well ordering, uniformization, and scale) hold in the dual (opposite) classes.
- 6.
Deep research by several set theorists including Martin, Steel, Kechris, Foreman, Magidor, Shelah, and Woodin, culminated in the final ideas and results.
- 7.
Woodin showed that with a marginally stronger hypothesis (existence of a measurable cardinal above infinitely many Woodin cardinals) the determinacy of a much larger class of sets (than the projective sets) called \(L(\mathbf{R})\) can be established.
- 8.
This was shown by Cohen, Levy, and Solovay.
- 9.
The web site http://logic.harvard.edu/efi.php has more information and resources. The project is funded by a grant from the John Templeton Foundation.
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Dasgupta, A. (2014). Postscript IV: Landmarks of Modern Set Theory. In: Set Theory. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-8854-5_22
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