For Mathematicians: The Main Idea

In the history of set theory, it became apparent to many researchers in the late 1960s that, a characteristic feature of bigger large cardinals is that they arise as critical points of elementary embeddings of the universe. These are typically of the form
j : V -> M, where M is a transitive model of ZFC that is a proper subclass of V. Moreover, the stronger the large cardinal notion, the more closely M is seen to resemble V itself. The natural question in those early days was: Could M = V?

Soon after such questions arose, Kunen showed that there could be no elementary embedding from V to itself other than the identity. Since the statement could not be formulated in the language of ZFC, Kunen pointed out that his proof must be formalized in an appropriate class theory, such as Kelley-Morse or Godel-Bernays class theory.

The author of The Infinite According to Master Wu, as a young set theorist, became convinced that there "ought" to be a nontrivial elementary embedding from V to itself. His conviction arose from the natural parallels he found between the structure of V and the structure of "wholeness" or "Absolute Infinte" according to certain ancient visions of infinity. Looking more closely at Kunen's work, the author noticed that Kunen's proof actually depends on the context in which it is formulated -- in the standard class theories, any elementary embedding from V to itself ends up being (weakly) definable, which forces it to have additional properties beyond elementarity. Indeed, it is these additional properites in combination with elementarity that lead to inconsistency, not the elementarity of the embedding on its own.

The author formulated a context for studying elementary embeddings from V to itself in which one could axiomatically decide exactly which properties, in addition to elementarity, an embedding from V to itself would have. In this context, the author showed that many different large cardinal strengths arise when the axiomatic assumptions imposed on the embedding vary, with inconsistency arising as an extreme special case. See a description of the main research papers here.

The author proposed, in several articles, a new axiom schema, which he called the Wholeness Axiom, intended to be added to the usual axioms of ZFC (in the expanded language described above) as a way of providing an axiomatic account for virtually all large cardinals. He found that inclusion of the Wholeness Axiom among the foundational axioms revealed much deeper connections to the view of the Infinite in ancient texts. The ancient texts suggested certain dynamics within the Infinite; natural analogues to these dynamics became apparent once the Wholeness Axiom was assumed.

The author also found through experience that many of the ideas related to the Wholeness Axiom were accessible to non-mathematical audiences, especially when the analogies found in the ancient texts were included in the presentation of the subject. This observation inspired the author to provide an accessible treatment of these ideas, and some of their practical application, in the form of a book.