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On needed reals

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Abstract

Given a binary relationR, we call a subsetA of the range ofR R-adequate if for everyx in the domain there is someyεA such that (x, y)εR. Following Blass [4], we call a realη ”needed” forR if in everyR-adequate set we find an element from whichη is Turing computable. We show that every real needed for inclusion on the Lebesgue null sets,Cof(\(\mathcal{N}\)), is hyperarithmetic. Replacing “R-adequate” by “R-adequate with minimal cardinality” we get the related notion of being “weakly needed”. We show that it is consistent that the two notions do not coincide for the reaping relation. (They coincide in many models.) We show that not all hyperarithmetic reals are needed for the reaping relation. This answers some questions asked by Blass at the Oberwolfach conference in December 1999 and in [4].

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References

  1. U. Abraham,Proper forcing, inHandbook of Set Theory (M. Foreman, A. Kanamori and M. Magidor, eds.), Kluwer Academic Publishers, Dordrecht, to appear.

  2. T. Bartoszyński and H. Judah,Set Theory, On the Structure of the Real Line, A K Peters, Wellesley, Massachusetts, 1995.

    Google Scholar 

  3. A. Blass,Needed reals, Talk at Oberwolfach, December 1999.

  4. A. Blass,Needed reals and recursion in generic reals, Annals of Pure and Applied Logic109 (2001), 77–88.

    Article  MATH  MathSciNet  Google Scholar 

  5. K. de Leeuw, E. F. Moore, C. E. Shannon and N. Shapiro,Computability by probabilistic machines, inAutomata Studies (J. McCarthy and C. E. Shannon, eds.), Princeton University Press, 1956, pp. 183–212.

  6. M. Goldstern,A taste of proper forcing, inSet Theory (C. Augusto di Prisco, J. A. Larson, J. Bagaria and A. R. D. Mathias, eds.), Kluwer Academic Publishers, Dordrecht, 1998, pp. 71–82.

    Google Scholar 

  7. T. Jech,Set Theory, Addison Wesley, Reading, MA, 1978.

    Google Scholar 

  8. C. G. Jockusch Jr,Uniformly introreducible sets, Journal of Symbolic Logic33 (1968), 521–536.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. Oxtoby,Measure and Category, second edition, Springer, Berlin, 1980.

    MATH  Google Scholar 

  10. S. Shelah,Long iterations for the continuum, Preprint [Sh:707], 2000.

  11. R. Solovay,Hyperarithmetically computable sets, Transactions of the American Mathematical Society239 (1978), 99–122.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institut für formale Logik, Universität Wien, Währinger Str. 25, 1090, Wien, Austria

    Heike Mildenberger

  2. Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904, Jerusalem, Israel

    Saharon Shelah

Authors
  1. Heike Mildenberger
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  2. Saharon Shelah
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Corresponding author

Correspondence to Heike Mildenberger.

Additional information

The first author was supported by a Minerva fellowship and by grant 13983 of the Austrian Fonds zur wissenschaftlichen Förderung.

The second author’s research was partially supported by the “Israel Science Foundation”, founded by the Israel Academy of Science and Humanities. This is the second author’s work number 725.

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Mildenberger, H., Shelah, S. On needed reals. Isr. J. Math. 141, 1–37 (2004). https://doi.org/10.1007/BF02772209

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  • DOI: https://doi.org/10.1007/BF02772209

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