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# THE RELATION BETWEEN TWO DIMINISHED CHOICE PRINCIPLES

Part of: Set theory

Published online by Cambridge University Press:  15 February 2021

Corresponding

## Abstract

For every $n\in \omega \setminus \{0,1\}$ we introduce the following weak choice principle: $\operatorname {nC}_{<\aleph _0}^-:$ For every infinite family $\mathcal {F}$ of finite sets of size at least n there is an infinite subfamily $\mathcal {G}\subseteq \mathcal {F}$ with a selection function $f:\mathcal {G}\to \left [\bigcup \mathcal {G}\right ]^n$ such that $f(F)\in [F]^n$ for all $F\in \mathcal {G}$ .

Moreover, we consider the following choice principle: $\operatorname {KWF}^-:$ For every infinite family $\mathcal {F}$ of finite sets of size at least $2$ there is an infinite subfamily $\mathcal {G}\subseteq \mathcal {F}$ with a Kinna–Wagner selection function. That is, there is a function $g\colon \mathcal {G}\to \mathcal {P}\left (\bigcup \mathcal {G}\right )$ with $\emptyset \not =f(F)\subsetneq F$ for every $F\in \mathcal {G}$ .

We will discuss the relations between these two choice principles and their relations to other well-known weak choice principles. Moreover, we will discuss what happens when we replace $\mathcal {F}$ by a linearly ordered or a well-ordered family.

## MSC classification

Type
Article
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The Journal of Symbolic Logic , March 2021 , pp. 415 - 432
© The Association for Symbolic Logic 2021

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