TITLE: Reinhardt's ultimate classes
QUESTION [5 upvotes]: 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.

REPLY [5 votes]: You can find Reinhardt's philosophy of set theory in

"Set existence principles of Shoenfield, Ackermann, and Powell", Fundamenta Mathematica, vol 84, pp 5-34 and
"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!