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CANTOR’S THEOREM MAY FAIL FOR FINITARY PARTITIONS

Part of: Set theory

Published online by Cambridge University Press:  03 April 2024

GUOZHEN SHEN Open the ORCID record for GUOZHEN SHEN [Opens in a new window]
GUOZHEN SHEN*
Affiliation:
SCHOOL OF PHILOSOPHY WUHAN UNIVERSITY NO. 299 BAYI ROAD, WUHAN, HUBEI PROVINCE 430072 PEOPLE’S REPUBLIC OF CHINA
*
E-mail: [email protected]
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Abstract

A partition is finitary if all its members are finite. For a set A, $\mathscr {B}(A)$ denotes the set of all finitary partitions of A. It is shown consistent with $\mathsf {ZF}$ (without the axiom of choice) that there exist an infinite set A and a surjection from A onto $\mathscr {B}(A)$. On the other hand, we prove in $\mathsf {ZF}$ some theorems concerning $\mathscr {B}(A)$ for infinite sets A, among which are the following:

  1. (1) If there is a finitary partition of A without singleton blocks, then there are no surjections from A onto $\mathscr {B}(A)$ and no finite-to-one functions from $\mathscr {B}(A)$ to A.

  2. (2) For all $n\in \omega $, $|A^n|<|\mathscr {B}(A)|$.

  3. (3) $|\mathscr {B}(A)|\neq |\mathrm {seq}(A)|$, where $\mathrm {seq}(A)$ is the set of all finite sequences of elements of A.

Keywords

Cantor’s theoremaxiom of choicefinitary partitionpermutation model

MSC classification

Primary: 03E10: Ordinal and cardinal numbers 03E30: Axiomatics of classical set theory and its fragments
Secondary: 03E25: Axiom of choice and related propositions 03E35: Consistency and independence results
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 18
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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