We will prove that there exists a model of ZFC+``c= omega_2'' in which every
M subseteq R of cardinality less than continuum c is meager, and such that for
every X subseteq R of cardinality c there exists a continuous function f:R-> R
with f[X]=[0,1]. In particular in this model there is no magic set, i.e., a set
M subseteq R such that the equation f[M]=g[M] implies f=g for every continuous
nowhere constant functions f,g:R-> R 

Ciesielski, Krzysztof

Shelah, Saharon

English

arXiv

We will prove that there exists a model of ZFC+“c = ω2 ” in which every M ⊆ R of cardinality less than continuum c is meager, and such that for every X ⊆ R of cardinality c there exists a continuous function f: R → R with f [X]  = [0, 1]. In particular in this model there is no magic set, i.e., a set M ⊆ R such that the equation f [M]  = g[M] implies f = g for every continu-ous nowhere constant functions f, g: R → R. ∗The authors wish to thank Professors Maxim Burke, Andrzej RosClanowski and Jerzy Wojciechowski for reading preliminary versions of this paper and helping in improving its final version

A model with no magic sets

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