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A model with no magic set

Published online by Cambridge University Press:  12 March 2014

Krzysztof Ciesielski
Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, USA, E-mail:
Saharon Shelah
Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, USA, E-mail:


We will prove that there exists a model of ZFC+“c =” 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 continuous nowhere constant functions f,g: ℝ → ℝ.

Research Article
Copyright © Association for Symbolic Logic 1999

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