# Consequences of arithmetic for set theory

Journal of Symbolic Logic 59 (1):30-40 (1994)

# 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 C and D, either C ≤ D or D ≤ C. However, in ZF this is no longer so. For a given infinite set A consider $\operatorname{seq}^{1 - 1}(A)$ , the set of all sequences of A without repetition. We compare $|\operatorname{seq}^{1 - 1}(A)|$ , the cardinality of this set, to |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 $ZF \vdash \forall A(|\operatorname{seq}^{1 - 1}(A)| \neq|\mathscr{P}(\mathscr{A})|)$ , and we show that this is the best possible result. Furthermore, it is provable in ZF that if B is an infinite set, then $|\operatorname{fin}(B)| <|\mathscr{P}(B)|$ even though the existence for some infinite set B* of a function f from $\operatorname{fin}(B^\ast)$ onto P(B*) is consistent with ZF

# Links

## PhilArchive

Upload a copy of this work     Papers currently archived: 76,479

## External links

Setup an account with your affiliations in order to access resources via your University's proxy server

# Similar books and articles

E pluribus unum: Plural logic and set theory.John P. Burgess - 2004 - Philosophia Mathematica 12 (3):193-221.
Stretchings.O. Finkel & J. P. Ressayre - 1996 - Journal of Symbolic Logic 61 (2):563-585.
On interpretations of bounded arithmetic and bounded set theory.Richard Pettigrew - 2009 - Notre Dame Journal of Formal Logic 50 (2):141-152.
Mapping a set of reals onto the reals.Arnold W. Miller - 1983 - Journal of Symbolic Logic 48 (3):575-584.
Set theoretic properties of Loeb measure.Arnold W. Miller - 1990 - Journal of Symbolic Logic 55 (3):1022-1036.

# Analytics

Added to PP
2009-01-28

Downloads
51 (#232,844)

6 months
1 (#455,463)

Historical graph of downloads
How can I increase my downloads?

# Citations of this work

A weird relation between two cardinals.Lorenz Halbeisen - 2018 - Archive for Mathematical Logic 57 (5-6):593-599.
Factorials and the finite sequences of sets.Nattapon Sonpanow & Pimpen Vejjajiva - 2019 - Mathematical Logic Quarterly 65 (1):116-120.
A Note on Strongly Almost Disjoint Families.Guozhen Shen - 2020 - Notre Dame Journal of Formal Logic 61 (2):227-231.

# References found in this work

Set Theory.Thomas Jech - 1999 - Studia Logica 63 (2):300-300.
Zur Axiomatik der Mengenlehre (Fundierungs‐ und Auswahlaxiom).Ernst Specker - 1957 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 3 (13‐20):173-210.
Zur Axiomatik der Mengenlehre.Ernst Specker - 1957 - Mathematical Logic Quarterly 3 (13-20):173-210.
The Axiom of Choice.Gershon Sageev - 1976 - Journal of Symbolic Logic 41 (4):784-785.