Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Search

Refine search

Refine search

Access:
Content type:
Publication date:
Apply   From and To filters
Subject: Show more
Journals Show more
Publishers: Show more
Societies Show more
Series: Show more
Collections: Show more

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

2 results

  • 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

  • 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.

Paradoxes and Inconsistent Mathematics
  • Zach Weber
  • Published online:
    08 October 2021
    Print publication:
    21 October 2021
  • Logical paradoxes – like the Liar, Russell's, and the Sorites – are notorious. But in Paradoxes and Inconsistent Mathematics, it is argued that they are only the noisiest of many. Contradictions arise in the everyday, from the smallest points to the widest boundaries. In this book, Zach Weber uses “dialetheic paraconsistency” – a formal framework where some contradictions can be true without absurdity – as the basis for developing this idea rigorously, from mathematical foundations up. In doing so, Weber directly addresses a longstanding open question: how much standard mathematics can paraconsistency capture? The guiding focus is on a more basic question, of why there are paradoxes. Details underscore a simple philosophical claim: that paradoxes are found in the ordinary, and that is what makes them so extraordinary.