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Relations Between Some Cardinals in the Absence of the Axiom of Choice

Published online by Cambridge University Press:  15 January 2014

Lorenz Halbeisen
Affiliation:
Department of Pure Mathematics, Queen's University Belfast, Belfast BT7 1NN, Northern Ireland Department of Mathematics, University of Californiaat Berkeley, Berkeley, California 94720, USAE-mail:halbeis@math.berkeley.edu
Saharon Shelah
Affiliation:
Institute of Mathematics, The Hebrew University Jerusalem, Jerusalem 91904, IsraelE-mail:shelah@math.huji.ac.il

Abstract

If we assume the axiom of choice, then every two cardinal numbers are comparable. In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in specific permutation models and give some results provable without using the axiom of choice.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2001

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