Journal of Symbolic Logic

A Model with No Magic Set

Krzysztof Ciesielski and Saharon Shelah

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

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J. Symbolic Logic, Volume 64, Issue 4 (1999), 1467-1490.

First available in Project Euclid: 6 July 2007

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Ciesielski, Krzysztof; Shelah, Saharon. A Model with No Magic Set. J. Symbolic Logic 64 (1999), no. 4, 1467--1490.

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