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A model with no magic set

Published online by Cambridge University Press:  12 March 2014

Krzysztof Ciesielski  and
Saharon Shelah
Krzysztof Ciesielski
Affiliation:
Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, USA, E-mail: KCies@wvnvms.wvnet.edu
Saharon Shelah
Affiliation:
Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, USA, E-mail: shelah@math.huji.ac.il
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Abstract

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 4 , December 1999 , pp. 1467 - 1490
Copyright
Copyright © Association for Symbolic Logic 1999

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References

REFERENCES

[BaJu] Bartoszyński, T. and Judah, H., Set theory, A K Peters, 1995.CrossRefGoogle Scholar
[BaLa] Baumgartner, J. and Laver, R., Iterated perfect-set forcing, Annals of Mathematical Logic, vol. 17 (1979), pp. 271–288.CrossRefGoogle Scholar
[BeDi] Berarducci, A. and Dikranjan, D., Uniformly approachable functions and UA spaces, Rend. 1st. Matematica Univ. di Trieste, vol. 25 (1993), pp. 23–56.Google Scholar
[BuCi] Burke, M. R. and Ciesielski, K., Sets on which measurable functions are determined by their range, Canadian Mathematical Journal, vol. 49 (1997), pp. 1089–1116, (Preprint* available).CrossRefGoogle Scholar
[BuCi2] Burke, M. R. and Ciesielski, K., Sets of range uniqueness for classes of continuous functions, Proceedings of the American Mathematical Society (Preprint* available2), Posted on 05 11, 1999, PII:S0002-9939(99) 04905-9, to appear in print.Google Scholar
[Ci] Ciesielski, K., Set theory for the working mathematician, London Mathematical Society Student Texts, vol. 39, Cambridge University Press, 1997.CrossRefGoogle Scholar
[Co] Corazza, P., The generalized Borel conjecture and strongly proper orders, Transactions of the American Mathematical Society, vol. 316 (1989), pp. 115–140.CrossRefGoogle Scholar
[Ku] Kunen, K., Set theory, North-Holland, 1983.Google Scholar
[Mi] Miller, A. W., Mapping a set of reals onto the reals, this Journal, vol. 48 (1983), pp. 575–584.Google Scholar
[RoSh470] Rosłanowski, A. and Shelah, S., Norms on possibilities I: forcing with trees and creatures, Memoirs of the American Mathematical Society, in print.Google Scholar
[Sh] Shelah, S., Proper forcing, Lectures Notes in Mathematics, vol. 940, Springer-Verlag, 1982.CrossRefGoogle Scholar