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I always had the dream to design a course for my graduate students like "mathematical models of the continuum". This course should cover history of real numbers, the Measure Problem, the Banach-Tarski Paradox, but also Baire/Cantor/Polish spaces, and it should look at infinite games and determinacy as well.

Obviously I am conscious about the text books and articles covering all these concepts. However, I seem to be unable to find a rigorours, graduate-level monograph that captures all of this in one, which I could potentially use as a text.

Would someone be able to point me to a good reference for that?

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    $\begingroup$ You might be interested in the topic of descriptive set theory, which treats many of the topics you seem to have in mind. $\endgroup$
    – Wojowu
    Jul 22, 2020 at 17:57

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I like your idea of such a course a lot! If it is appropriate to recommend a book in German language, I think this one could be the perfect match:

Oliver Deiser (2007): Reelle Zahlen: Das klassische Kontinuum und die natürlichen Folgen

I own this book and can say it covers all the topics that you mentioned, and it is certainly a graduate-level text. I find the quality of the exposition outstanding, and the range of topics quite unique. It covers the historical development quite extensively and gives many references, with a focus on the original sources.

I think there is also a 2nd edition from 2008.

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A great textbook for your course would be "The Structure of the Real Line" by Lev Bukovský. It covers all of the topics you mentioned, except for the Banach-Tarski Paradox, and provides all necessary topological and measure-theoretic background.

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    $\begingroup$ I was just going to suggest this one, Santi. $\endgroup$ Aug 10, 2020 at 13:42
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    $\begingroup$ I finally got around to getting a copy of this book (arrived a few days ago). This is probably the most useful book (in English, known to me at this time) to have on hand for what the OP has in mind, although it may be a bit too encyclopedic to be useful as an actual text, at least as the primary text. If someone wanted a short list of books in English to pursue after Dasgupta's book, I would definitely include Bukovský's and Krechis's books in my list. $\endgroup$ Oct 12, 2020 at 9:51
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For english references:

An history of mathematics book, but extremely well written and mathematically sophisticated, with tons of references (that might be useuful) that adress all such things is

  • G. H. Moore's Zermelo's Axiom of Choice: Its Origins, Development, and Influence.

This should be my top pic. Other sources I know of:

  • D. L. Cohn's Measure Theory.

has a very nice introduction to Polish spaces and analytic sets, and

  • Folland's Real Analysis: Modern Techniques and Their Applications.

discusses the measure problem and Banach-Tarksi's Theorem, and has plenty references.

However, the level is more of upper-undergraduate then graduate, I think.

For descriptive set theory we have Krechis Classical descriptive set theory and Moschovakis Descriptive Set theory, but I guess that, by what you've said you know their content already.

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Maybe Set Theory. With an Introduction to Real Point Sets by Abhijit Dasgupta. See the chapter and section titles at amazon.com, especially the section titles, which indicate much better than just the chapter titles in showing the topics included. However, the book is probably better described as an "advanced undergraduate / beginning graduate" level text (U.S. standards) than a true graduate-level monograph.

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I can recommend two books to you which I think give a rather good coverage of the foundations for determinacy and infinite games

Hugh Woodin: The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal. Walter de Gruyter, Berlin 1999.

Gerald Sacks (ed.): Mathematical Logic in the 20th Century, SUP and World Scientific Publishing Singapore and London, 2003 (note especially the article from Donald Martin on Analytic Games) https://www.worldscientific.com/doi/pdf/10.1142/9789812564894_fmatter

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  • $\begingroup$ I know Woodin's monograph and can say it is excellent. I would say clearly upper graduate / research level. Maybe an important consideration for OP, one of the main results is Woodin gives a canonical model in which the Continuum Hypothesis is FALSE. $\endgroup$
    – MaryS.
    Jul 31, 2020 at 7:54

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