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Long ago, while I was reading a book ($*$) about the various way to build set theories (Zermelo-Freankel, Von Neumann–Bernays–Gödel, and type theories), I read about a variant of type theory with weird metalogics properties called Hao Wang's $\mathfrak S$ system, and presented for the first time in the year 1954.

The weird properties were something about his ability to avoid the Goedel's theorems.

Now I can't find that book anymore, and on internet I can't find nothing about this. Someone can explain me in easy words that theory's special properties and help me finding some free readings and introductions about?

Edit: The thory allowed the types of the variables to have transfinite value like $\omega$ and can be extended to even bigger ordinals.

Update 05-28-13 I remember the title of the chapter, and was about "Cumulative orders and cumulative types".

($*$)The book was Ettore Casari - Questioni di filosofia della matematica. Milan: Feltrinelli 1964

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    $\begingroup$ I have to wonder about your choice for ?! at the end of the title. $\endgroup$
    – Asaf Karagila
    Commented May 12, 2013 at 9:48
  • $\begingroup$ @AsafKaragila I used "!" because I'm surprised..maybe I remember bad, but really I was not able to find anything. $\endgroup$
    – MphLee
    Commented May 12, 2013 at 9:51
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    $\begingroup$ If you can find it useful, I can give you the actual references (there are a couple of alternatives). For the content, it is mainly a constructive approach, but I cannot get really into the details simply because I don't know them. However, as far as I can see, the way in which the system avoids the Godel's theorems is not - so to speak - straight (simply cause you cannot avoid them). This paper is more philosophical than technical, and it is about the formalization of mathematics in general. $\endgroup$
    – Kolmin
    Commented May 28, 2013 at 10:29
  • $\begingroup$ @Kolmin Interesting! You can post it as a commentif you want. $\endgroup$
    – MphLee
    Commented May 28, 2013 at 11:11
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    $\begingroup$ Ops... I read your comment too late. In the meantime I was writing an answer cause I had found quite some info (beyond the comment space). Again, I hope it will help. $\endgroup$
    – Kolmin
    Commented May 28, 2013 at 11:17

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The following is a list of the papers written by Hao Wang on the topic and published on journals:

1) The Formalization of Mathematics, Journal of Symbolic Logic 19, 241-266;
2) Ordinals Number and Predicative Set Theories, Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 5, 216-239.

There is another paper that has the title "Some formal details on Predicative Set Theories" (3). Interestingly, all those papers can be found in the following Hao Wang's book:

  • A Survey of Mathematical Logic, Amsterdam: North Holland, 1963.
    [Reprinted as Logic, Computers and Sets, New York: Chealsea, 1970]

In particular, we have that paper (1) (the introductory one and the most philosophical, with an interesting introduction on the method of mathematics) is on chapter XXIII [pp.559-584], paper (3) is on chapter XXIV [pp.585-623], and paper (2) is on chapter XXV [pp.624-651].

Last piece of information about a presentation of the theory, that is about introducing the constructive approach in mathematics, is this review by Mueller (unfortunately, the other two pages are not visible, but they can be easily downloaded with a JSTOR account).

For the sake of completeness, it is possible to find some information in the "Handbook of Philosophy of Mathematics" edited by Andrew Irvine in the part on Constructivism in Mathematics under the section Predicativsm [pp.333-337].

I hope it helps.

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  • $\begingroup$ Thanks, do you know if there is something online too? $\endgroup$
    – MphLee
    Commented May 28, 2013 at 13:37
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    $\begingroup$ As far as I know, online there is nothing in the form of free lectures, slides or notes on the topic. It seems that it is interesting more for historical reasons that as a starting point of something. However, considering that I assume you are an intalian speaker, there is the possibility that Carlo Cellucci (former student of Casari) worked on something like this in the years between his work on mathematical logic, and his shift towards philosophy of mathematics, in particular given the strong influence of Dummett and his constructive approach to the italian community of those years. $\endgroup$
    – Kolmin
    Commented May 28, 2013 at 13:47
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    $\begingroup$ However, I wouldn't bet my life on this. Anyway, good luck. I hope this hint will be fruitful. $\endgroup$
    – Kolmin
    Commented May 28, 2013 at 13:48
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Wang's system is discussed in some detail in "Foundations of set theory" by Fraenkel, Bar-Hillel and Levy (in Studies in logic and foundations of mathematics, vol. 67, pp. 175--178). The reference is to the "The Formalization of Mathematics" paper cited by above by Kolmin.

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