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1 - Introduction
Sharon Berry
Book:

A Logical Foundation for Potentialist Set Theory

Published online:

10 February 2022

Print publication:

17 February 2022, pp 1-10
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Summary

This chapter introduces the book’s key aims: showing how a logical possibility operator helps formulate potentialist set theory, intuitively justify the ZFC axioms, clarify applied mathematics and more.

2 - Actualist Set Theory
from Part I
Sharon Berry
Book:

A Logical Foundation for Potentialist Set Theory

Published online:

10 February 2022

Print publication:

17 February 2022, pp 13-30
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Summary

This chapter discusses problems for actualist concerning the intended height of the hierarchy of sets and intuitively justifying the axiom of Replacement.

5 - Set Theory
from Part III - Where Are the Paradoxes?
Zach Weber, University of Otago, New Zealand
Book:

Paradoxes and Inconsistent Mathematics

Published online:

08 October 2021

Print publication:

21 October 2021, pp 151-188
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Summary

In this chapter, the basic theory of sets is developed axiomaticallyin a paraconsistent logic. The two main goals are (1) to establish atoolkit for elementary mathematics, and (2) to prove the mainantinomies of naive set theory. The two goals come together inproving the Burali-Forti paradox for the theory of ordinals. Alongthe way, results are proved about the universal set, various formsof “empty” sets, Russell’s set, the axioms ofZFC, fixed points, Cantor’s theorem, and the possibility of awell-ordering theorem. The Routley set is introduced and studied asa particularly inconsistent object.