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A NEW WEAK CHOICE PRINCIPLE
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- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 19 February 2026, pp. 1-18
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For every natural number n we introduce a new weak choice principle
$\mathrm {nRC_{fin}}$:Given any infinite set x, there is an infinite subset
$y\subseteq x$ and a selection function f that chooses an n-element subset from every finite 
$z\subseteq y$ 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
$\mathrm {nRC_{fin}}$ to the weak choice principles 
$\mathrm {RC_m}$ (that has already been studied in [3] and [6]):Given any infinite set x, there is an infinite subset
$y\subseteq x$ with a choice function f on the family of all m-element subsets of y.Moreover, we prove a stronger analogue of the results in [6] when we study the relation between
$\mathrm {nRC_{fin}}$ and 
$\mathrm {kC_{fin}^-}$ which is defined byGiven any infinite family
$\mathcal {F}$ of finite sets of cardinality greater than k, there is an infinite subfamily 
$\mathcal {A}\subseteq \mathcal {F}$ with a selection function f that chooses a k-element subset from each 
$A\in \mathcal {A}$.
The theory of functional grammar. Part 1: The structure of the clause. 2nd edn. By Simon C. Dik (Functional grammar series 20.) Berlin: Mouton de Gruyter, 1997. Pp. xx, 509. Part 2: Complex and derived constructions. 1st edn. By Simon C. Dik. Ed. By Kees Hengeveld. (Functional grammar series 21.) Berlin: Mouton de Gruyter, 1997. Pp. xx, 477.
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