1 results
CANTOR’S THEOREM MAY FAIL FOR FINITARY PARTITIONS
- Part of
-
- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 03 April 2024, pp. 1-18
-
- Article
- Export citation
-
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) 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) For all
$n\in \omega $, 
$|A^n|<|\mathscr {B}(A)|$.(3)
$|\mathscr {B}(A)|\neq |\mathrm {seq}(A)|$, where 
$\mathrm {seq}(A)$ is the set of all finite sequences of elements of A.
