Abstract
It is a commonplace of set theory to say that there is no set of all well-orderings nor a set of all sets. We are implored to accept this due to the threat of paradox and the ensuing descent into unintelligibility. In the absence of promising alternatives, we tend to take up a conservative stance and tow the line: there is no universe [Halmos, 1960]. In this paper, I am going to challenge this claim by taking seriously the idea that we can talk about the collection of all the sets and many more collections beyond that. A method of articulating this idea is offered through an indefinitely extending hierarchy of set theories. It is argued that this approach provides a natural extension to ordinary set theory and leaves ordinary mathematical practice untouched.
| Original language | English |
|---|---|
| Pages (from-to) | 1875-1907 |
| Number of pages | 33 |
| Journal | Synthese |
| Volume | 193 |
| Issue number | 6 |
| Early online date | 8 Jul 2015 |
| DOIs | |
| Publication status | Published - Jun 2016 |
Bibliographical note
I would like to thank Zach Weber, Øystein Linnebo, Carrie Jenkins, James Studd, Stephen Read, Volker Halbach, Jc Beall, Dan Isaacson, Torfinn Huvenes and Kentaro Fujimoto for providing invaluable assistance in the development of this paper. I would also like to thank Oxford University and the University of St Andrews for giving me the opportunity to present these ideas. Finally, I would like to thank two anonymous referees for their incisive comments and suggestions for the paper.Keywords
- Philosophy of mathematics
- Philosophy of set theory
- Formal theories of truth
- Absolute generality
- Indefinite extensibility