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In the preface to Sets and Classes by Muller, several research programs are outlined that were in development concurrently with publication (or finished slightly beforehand) that he would have liked to discuss, but was unable to for various reasons.

One such program is called ultimate classes, apparently an attempt to explore the outer limits of extensions of $ZF$ via axiomatic projection schemata by adding a new primitive concept to set theory (in addition to $\in$) corresponding to embeddings $\mathbb{V}\to\mathbb{V}$ of the universe into itself.

Apparently, if consistent these schemata provide for the existence of extendible cardinals of high degree in addition to many other large cardinals.

Where can I read about Reinhardt's work on this approach?

Any relevant pointers are greatly appreciated.

In response to a comment by Matt F., here is the relevant preface from Muller's 1976 book.

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  • $\begingroup$ Can you provide a reference and a date for the preface that you mention? I see a book titled Sets and Classes edited by Gert Muller in 1976, and a paper titled Sets, Classes and Categories by F.A. Mueller in 2000. The lack of a date is especially awkward given the comment on programs in development at the time of publication. $\endgroup$
    – user44143
    Commented Mar 12, 2023 at 2:49
  • $\begingroup$ @MattF. Reinhardt passed in '98 and wasn't releasing any nachlass to my knowledge, so without looking I'm confident it's the 1976 book -- for a reference you can see here, however you'll need institutional access for the preface (mine is a hard copy, I might be able to find it and take photos but I think it's elsewhere at the moment). $\endgroup$
    – Alec Rhea
    Commented Mar 13, 2023 at 14:28

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You can find Reinhardt's philosophy of set theory in

  1. "Set existence principles of Shoenfield, Ackermann, and Powell", Fundamenta Mathematica, vol 84, pp 5-34 and
  2. "Remarks on reflection principles, large cardinals, and elementary embeddings", Proceedings of symposia in pure mathematics, vol 13, part 2, American Mathematical Society, Providence 1974, pp 189-205. Downloadable pdf in the Math Stack Exchange question, "How does Reinhardt's extension of the set-theoretical universe beyond $V_\Omega$ work?"

Compare and contrast Reinhardt's philosophy of set theory with the philosophy of set theory implicit in Kunen's paper introducing the "Kunen Inconsistency". Happy Hunting!

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    $\begingroup$ Alec, also see Reinhardt's "Ackermann's set theory equals ZF", Ann of Math Logic 2 (1970), pp. 189-249. $\endgroup$
    – Philip Ehrlich
    Commented May 7, 2022 at 4:05
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    $\begingroup$ (The PDF referenced in point 2 has a better scan as part of the libgen copy of "Axiomatic Set Theory, Part 2" (which is wrongly listed as Part 1). I shall refrain from posting the link) $\endgroup$
    – Calvin Khor
    Commented May 7, 2022 at 6:00

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