# Sh:488

• Halbeisen, L. J., & Shelah, S. (1994). Consequences of arithmetic for set theory. J. Symbolic Logic, 59(1), 30–40.
• 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 \mathcal{C} and \mathcal{D}, either \mathcal{C} \leq \mathcal{D} or \mathcal{D} \leq \mathcal{C}. However, in ZF this is no longer so. For a given infinite set A consider Seq (A), the set of all sequences of A without repetition. We compare |Seq (A)|, the cardinality of this set, to |{\mathcal {P}}(A)|, 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\vdash\forall A: |Seq(A)|\neq|{\mathcal{P}}(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)|<|{\mathcal{P}}(B)|, even though the existence for some infinite set B^* of a function f from fin(B^*) onto {\mathcal{P}}(B^*) is consistent with ZF.

• Version 1993-08-27_10 (14p) published version (12p)
Bib entry
@article{Sh:488,
author = {Halbeisen, Lorenz J. and Shelah, Saharon},
title = {{Consequences of arithmetic for set theory}},
journal = {J. Symbolic Logic},
fjournal = {The Journal of Symbolic Logic},
volume = {59},
number = {1},
year = {1994},
pages = {30--40},
issn = {0022-4812},
mrnumber = {1264961},
mrclass = {03E35 (03E10 03E25)},
doi = {10.2307/2275247},
note = {\href{https://arxiv.org/abs/math/9308220}{arXiv: math/9308220}},
arxiv_number = {math/9308220}
}