# Sh:488

- Halbeisen, L. J., & Shelah, S. (1994).
*Consequences of arithmetic for set theory*. J. Symbolic Logic,**59**(1), 30–40. arXiv: math/9308220 DOI: 10.2307/2275247 MR: 1264961 -
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} }