A Logical Foundation for Potentialist Set Theory


[Draft Chapters: Any comments on style and/or content to would be much appreciated!][Chapters 3-6 are now slightly revised for readability. Sincere thanks to A.Y. for his input.]


Abstract:

Potentialism is a popular philosophy of set theory, which has been developed by philosophers like Putnam, Parsons and Hellman. It solves apparent paradoxes about the height of the hierarchy of sets by reinterpreting set theory as an exploration of how it would be logically or mathematically possible for standard-width initial segments of the hierarchy of sets to be extended.

In this book, I introduce a powerful, yet intuitively compelling, formal system for reasoning about logical possibility. I then use this formal system to address a major line of worry for potentialism: that wholeheartedly adopting potentialism makes current mathematical practice look unjustified. Specifically, I show that every mainstream (ZFC based) proof in set theory can be transformed into a proof of the potentialist translation for the relevant claim within my formal system. This result also turns out to provide a new and appealing answer to Boolos' classic question about how anyone (realist or potentialist) can justify use of the axiom of replacement.


Table of contents


Chapters:
  1. Introduction
  2. Philosophical Background and the Problem to be Solved
  3. The Language of Logical Possibility ℒ
  4. The Formal System I: Basics
  5. The Formal System II: Other Inference Rules
  6. Example Lemmas About Well-Ordering
  7. Useful Corollaries to Axioms
  8. Inf. Lemma & Wrapping Trick
  9. The Potentialist Translation
  10. Use of First Order Logic in Set Theory & Translation Lemma [note: these last two chapters heavily use the Wrapping Trick from ch8, so maybe read that first?]
  11. Use of ZFC in Set Theory

Appendixes: [coming soon!]
  1. Set Theoretic Mimicry of ℒ
  2. Recursive Definition and Isomorphism Lemmas
  3. Helpful Facts About Hierarchies