The Future of Post-Human Set Theory
Two Volume Set

Book Abstract

Is set theory really so foundational in mathematics that David Hilbert once said that “no one shall expel us from the paradise which Cantor has created for us,” when he expressed the importance of Georg Cantor’s set theory in the development of mathematics? (GO 2016) This positive view on set theory can be contrasted with a rebuke by Richard Hamming, who rhetorically said: “I know that the great Hilbert said ‘We will not be driven out of the paradise Cantor has created for us,’ and I reply ‘I see no reason for walking in!’” (ST 2016)

Contrary to these opposing views (and other ones as will be discussed in the book), set theory (in relation to function and non-function) is neither possible (or impossible) nor desirable (or undesirable) to the extent that the respective ideologues (on different sides) would like us to believe.

This challenge to the conventional wisdom in set theory does not mean that it is worthless, or that those diverse fields (related to set theory)—like arithmetic, algebra, mathematical logic, category theory, type theory, geometry, calculus, computation, model theory, philosophy of mathematics, systems engineering, evolutionary theory, aesthetics, meta-mathematics, and so on—should be ignored. In fact, neither of these extreme views is reasonable.

Instead, this book offers an alternative (better) way to understand the future of set theory (and related fields) in regard to the dialectic relationship between function and non-function—while learning from different approaches in the literature but without favoring any one of them (nor integrating them, since they are not necessarily compatible with each other). More specifically, this book offers a new theory (that is, the post-foundational theory of sets) to go beyond the existing approaches in a novel way and is organized in four chapters.

This seminal project will fundamentally change the way that we think about set theory (in relation to the dialectic relationship between function and non-function) from the combined perspectives of the mind, nature, society, and culture, with enormous implications for the human future and what I originally called its “post-human” fate.