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Unbounded and dominating reals in Hechler extensions

Published online by Cambridge University Press:  12 March 2014

Justin Palumbo
Justin Palumbo*
Affiliation:
Department of Mathematics, University of California, Los Angeles, California 90095-1555, USA, E-mail: [email protected]
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Abstract

We give results exploring the relationship between dominating and unbounded reals in Hechler extensions, as well as the relationships among the extensions themselves. We show that in the standard Hechler extension there is an unbounded real which is dominated by every dominating real, but that this fails to hold in the tree Hechler extension. We prove a representation theorem for dominating reals in the standard Hechler extension: every dominating real eventually dominates a sandwich composition of the Hechler real with two ground model reals that monotonically converge to infinity. We apply our results to negatively settle a conjecture of Brendle and Löwe (Conjecture 15 of [4]). We also answer a question due to Laflamme.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 1 , March 2013 , pp. 275 - 289
Copyright
Copyright © Association for Symbolic Logic 2013

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References

REFERENCES

[1]Bartoszyński, Tomek and Judah, Haim, Set theory, A K Peters Ltd., Wellesley, MA, 1995.CrossRefGoogle Scholar
[2]Baumgartner, James E. and Dordal, Peter, Adjoining dominating functions, this Journal, vol. 50 (1985), no. 1. pp. 94–101.Google Scholar
[3]Brendle, Jörg, Judah, Haim, and Shelah, Saharon, Combinatorial properties of Hechler forcing, Annals of Pure and Applied Logic, vol. 58 (1992), no. 3, pp. 185–199.CrossRefGoogle Scholar
[4]Brendle, Jörg and Löwe, Benedikt, Eventually different functions and inaccessible cardinals, Journal of the Mathematical Society of Japan, vol. 63 (2011), no. 1, pp. 137–151.CrossRefGoogle Scholar
[5]Groszek, Marcia J., Combinatorics on ideals and forcing with trees, this Journal, vol. 52 (1987), no. 3, pp. 582–593.Google Scholar
[6]Hamkins, Joel David (mathoverflow.net/users/1946), Cantor-Bernstein for notions of forcing, MathOverflow, http://mathoverflow.net/questions/79323 (version: 2011-10-29).Google Scholar
[7]Hechler, Stephen H., On the existence of certain cofinal subsets of ωΩ, Axiomatic set theory, Proceedings of Symposia in Pure Mathematics, vol. XIII, part II, American Mathematical Society, Providence, R.I., 1974, pp. 155–173.CrossRefGoogle Scholar
[8]Laflamme, Claude, Bounding and dominating number of families of functions on ω, Mathematical Logic Quarterly, vol. 40 (1994), no. 2, pp. 207–223.CrossRefGoogle Scholar
[9]Neeman, Itay, Optimal proofs of determinacy, The Bulletin of Symbolic Logic, vol. 1 (1995), no. 3, pp. 327–339.CrossRefGoogle Scholar
[10]Scheepers, Marion, Gaps in Ωω, Set theory of the reals, Israel Mathematical Conference Proceedings, vol. 6, Bar-Ilan University, Ramat Gan, 1993, pp. 439–561.Google Scholar
[11]Steel, J. R., Inner models with many Woodin cardinals, Annals of Pure and Applied Logic, vol. 65 (1993), no. 2, pp. 185–209.CrossRefGoogle Scholar
[12]Truss, John, Sets having calibre ℕ1, Logic Colloquium 76, Studies in Logic and the Foundations of Mathematics, vol. 87, North-Holland, Amsterdam, 1977, pp. 595–612.Google Scholar