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Cantor’s set theory from a modern point of view. (English) Zbl 1027.03038
This survey article gives a short description of modern developments in set thery. The author mentions that Cantor’s set theory provided no clear picture of the universe of sets as a whole and continues: “There is now evidence that such a picture is starting to emerge through the detailed analysis of subuniverses of the universe of all sets called inner models”. After some introductory remarks on Cantor’s work, in particular the continuum hypothesis, Gödel’s model \(L\) of ZFC and Cohen’s forcing method are discussed in short. The last section deals with the search for a canonical acceptable interpretation or standard model of ZFC which is larger than \(L\) and which provides “correct” answers to undecidable problems. An answer came from measure theory. Relevant results of Scott, Solovay and Silver are listed, and the role of large cardinal hypotheses is sketched – these calibrate the strength of natural set-theoretic assertions. Finally, the importance of Woodin cardinals and the problems around them are mentioned: The construction of canonical inner models for large cardinals at and above the level of Woodin cardinals is a central, ongoing project in pure set theory”.
Reviewer: E.Harzheim (Köln)
MSC:
03E15 Descriptive set theory
03-03 History of mathematical logic and foundations
03E35 Consistency and independence results
03E45 Inner models, including constructibility, ordinal definability, and core models
03E55 Large cardinals
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