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LEVEL THEORY, PART 3: A BOOLEAN ALGEBRA OF SETS ARRANGED IN WELL-ORDERED LEVELS

Published online by Cambridge University Press:  28 April 2021

TIM BUTTON*
Affiliation:
DEPARTMENT OF PHILOSOPHY UNIVERSITY COLLEGE LONDON GOWER STREET, LONDON, WC1E 6BT, UK E-mail: tim.button@ucl.ac.uk URL: http://www.nottub.com

Abstract

On a very natural conception of sets, every set has an absolute complement. The ordinary cumulative hierarchy dismisses this idea outright. But we can rectify this, whilst retaining classical logic. Indeed, we can develop a boolean algebra of sets arranged in well-ordered levels. I show this by presenting Boolean Level Theory, which fuses ordinary Level Theory (from Part 1) with ideas due to Thomas Forster, Alonzo Church, and Urs Oswald. BLT neatly implement Conway’s games and surreal numbers; and a natural extension of BLT is definitionally equivalent with ZF.

Type
Articles
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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