Consequences of arithmetic for set theory

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

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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.

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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.

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