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Consequences of arithmetic for set theory

Published online by Cambridge University Press:  12 March 2014

Lorenz Halbeisen
Affiliation:
Department of Mathematics, Eldgen. Technische Hochschule, Zürich, Switzerland, E-mail:halbeisen@math.ethz.ch
Saharon Shelah
Affiliation:
Institute of Mathematics, Hebrew University Jerusalem, Jerusalem., Israel, E-mail:shelah@math.huji.ac.il

Abstract

In this paper, we consider certain cardinals in ZF (set theory without AC, the axiom of choice). In ZFC (set theory with AC), given any cardinals and , either or . However, in ZF this is no longer so. For a given infinite set A consider seq1-1(A), the set of all sequences of A without repetition. We compare |seq1-1(A)|, the cardinality of this set, to ||, the cardinality of the power set of A. What is provable about these two cardinals in ZF? The main result of this paper is that ZF ⊢ ∀A(| seq1-1(A)| ≠ ||), and we show that this is the best possible result. Furthermore, it is provable in ZF that if B is an infinite set, then | fin(B)| < | (B*)| even though the existence for some infinite set B* of a function ƒ from fin(B*) onto (B*) is consistent with ZF.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1994

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References

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