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2015 Ordinal Exponentiations of Sets
Laurence Kirby
Notre Dame J. Formal Logic 56(3): 449-462 (2015). DOI: 10.1215/00294527-3132806


The “high school algebra” laws of exponentiation fail in the ordinal arithmetic of sets that generalizes the arithmetic of the von Neumann ordinals. The situation can be remedied by using an alternative arithmetic of sets, based on the Zermelo ordinals, where the high school laws hold. In fact the Zermelo arithmetic of sets is uniquely characterized by its satisfying the high school laws together with basic properties of addition and multiplication. We also show how in both arithmetics the behavior of exponentiation depends on whether the empty set is an element of the base.


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Laurence Kirby. "Ordinal Exponentiations of Sets." Notre Dame J. Formal Logic 56 (3) 449 - 462, 2015.


Received: 11 February 2013; Accepted: 30 March 2013; Published: 2015
First available in Project Euclid: 22 July 2015

zbMATH: 1334.03044
MathSciNet: MR3373613
Digital Object Identifier: 10.1215/00294527-3132806

Primary: 03E10 , 03E20

Keywords: exponentiation , ordinal arithmetic , Set theory

Rights: Copyright © 2015 University of Notre Dame


Vol.56 • No. 3 • 2015
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