THE RELATION BETWEEN TWO DIMINISHED CHOICE PRINCIPLES

Author:

SCHUMACHER SALOME

Abstract

AbstractFor 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.

Publisher

Cambridge University Press (CUP)

Subject

Logic,Philosophy

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