A New Weak Choice Principle

2021/01/19 by Halbeisen, Lorenz, Plati, Riccardo, Schumacher, Salome
#FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2101.07840

Abstract

For every natural number n we introduce a new weak choice principle \mathrmnRC_fin: Given any infinite set x, there is an infinite subset y⊆ x and a selection function f that chooses an n-element subset from every finite z⊆ y containing at least n elements. By constructing new permutation models built on a set of atoms obtained as Fraïssé limits, we will study the relation of \mathrmnRC_fin to the weak choice principles RC_m (that has already been studied by Montenegro, Halbeisen and Tachtsis): Given any infinite set x, there is an infinite subset y⊆ x with a choice function f on the family of all m-element subsets of y. Moreover, we prove a stronger analogue of Montenegros results when we study the relation between \mathrmnRC_fin and \mathrmkC_fin^- which is defined by: Given any infinite family F of finite sets of cardinality greater than k, there is an infinite subfamily A⊆ F with a selection function f that chooses a k-element subset from each A\inA.

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• Partial choice functions for families of finite sets
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