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-definability at uncountable regular cardinals

Published online by Cambridge University Press:  12 March 2014

Philipp Lücke
Philipp Lücke*
Affiliation:
Institut für Mathematische Logik und Grundlagenforschung, Fachbereich Mathematik und Informatik, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany, E-mail: [email protected]
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Abstract

Let κ be an infinite cardinal. A subset of (κκ)n is a -subset if it is the projection p[T] of all cofinal branches through a subtree T of (>κκ)n+1 of height κ. We define and -subsets of (κκ)n as usual.

Given an uncountable regular cardinal κ with κ = κ<κ and an arbitrary subset A of κκ, we show that there is a <κ-closed forcing ℙ that satisfies the κ+-chain condition and forces A to be a -subset of κκ in every ℙ-generic extension of V. We give some applications of this result and the methods used in its proof.

(i) Given any set x, we produce a partial order with the above properties that forces x to be an element of L.

(ii) We show that there is a partial order with the above properties forcing the existence of a well-ordering of κκ whose graph is a -subset of κκ × κκ.

(iii) We provide a short proof of a result due to Mekler and Väänänen by using the above forcing to add a tree T of cardinality and height κ such that T has no cofinal branches and every tree from the ground model of cardinality and height κ without a cofinal branch quasi-order embeds into T.

(iv) We will show that generic absoluteness for -formulae (i.e., formulae with parameters which define -subsets of κκ) under <κ-closed forcings that satisfy the κ+-chain condition is inconsistent.

In another direction, we use methods from the proofs of the above results to show that - and -subsets have some useful structural properties in certain ZFC-models.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 3 , September 2012 , pp. 1011 - 1046
Copyright
Copyright © Association for Symbolic Logic 2012

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References

REFERENCES

[1] Asperó, David and Friedman, Sy-David, Large cardinals and locally defined well-orders of the universe, Annals of Pure and Applied Logic, vol. 157 (2009), no. 1, pp. 1–15.Google Scholar
[2] Asperó, David and Friedman, Sy-David, Definable wellorderings of Hω2 and CH, Preprint available online at http://www.logic.univie.ac.at/~sdf/papers/joint.aspero.omega-2.pdf.Google Scholar
[3] Bagaria, Joan and Friedman, Sy D., Generic absoluteness, Proceedings of the XIth Latin American Symposium on Mathematical Logic (Mérida, 1998), vol. 108, 2001, pp. 3–13.Google Scholar
[4] Cummings, James, Iterated forcing and elementary embeddings, The handbook of set theory (Foreman, M., Kanamorie, A., and Magidor, M., editors), vol. 2, Springer, Berlin, 2010, pp. 775–884.CrossRefGoogle Scholar
[5] Feng, Qi, Magidor, Menachem, and Woodin, Hugh, Universally Baire sets of reals, Set theory of the continuum (Berkeley, CA, 1989), Mathematical Sciences Research Institute Publications, vol. 26, Springer, New York, 1992, pp. 203–242.CrossRefGoogle Scholar
[6] Friedman, Sy-David, Forcing, combinatorics and definability, Proceedings of the 2009 RIMS Workshop on Combinatorical Set Theory and Forcing Theory in Kyoto, Japan, RIMS Kokyuroku No. 1686, 2010, pp. 24–40.Google Scholar
[7] Friedman, Sy-David and Holy, Peter, Condensation and large cardinals, Fundamenta Mathematical vol. 215 (2011), no. 2, pp. 133–166.CrossRefGoogle Scholar
[8] Friedman, Sy-David, Hyttinen, Tapani, and Kulikov, Vadim, Generalised descriptive set theory and classification theory, Preprint available online at http://www.logic.univie.ac.at/~sdf/papers/joint.tapani.vadim.pdf.Google Scholar
[9] Fuchs, Gunter, Closed maximality principles: implications, separations and combinations, this Journal, vol. 73 (2008), no. 1, pp. 276–308.Google Scholar
[10] Harrington, Leo, Long projective wellorderings, Annals of Pure andApplied Logic, vol. 12 (1977), no. 1, pp. 1–24.Google Scholar
[11] Hyttinen, Tapani and Rautila, Mika, The canary tree revisited, this Journal, vol. 66 (2001), no. 4, pp. 1677–1694.Google Scholar
[12] Hyttinen, Tapani and Väänänen, Jouko, On Scott and Karp trees of uncountable models, this Journal, vol. 55 (1990), no. 3, pp. 897–908.Google Scholar
[13] Jech, Thomas J., Trees, this Journal, vol. 36 (1971), pp. 1–14.Google Scholar
[14] Jensen, R. B. and Solovay, R. M., Some applications of almost disjoint sets, Mathematical Logic and Foundations of Set Theory, North-Holland, Amsterdam, 1970, pp. 84–104.Google Scholar
[15] Kanamori, Akihiro, The higher infinite, second ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.Google Scholar
[16] Laver, Richard, Certain very large cardinals are not created in small forcing extensions, Annals of Pure and Applied Logic, vol. 149 (2007), no. 1–3, pp. 1–6.Google Scholar
[17] Mekler, Alan and Väänänen, Jouko, Trees and -subsets of ω1 ω1 , this Journal, vol. 58 (1993), no. 3, pp. 1052–1070.Google Scholar
[18] Nadel, Mark and Stavi, Jonathan, L∞λ,-equivalence, isomorphism and potential isomorphism, Transactions of the American Mathematical Society, vol. 236 (1978), pp. 51–74.Google Scholar
[19] Neeman, Itay and Zapletal, Jindřich, Proper forcings and absoluteness in L(R), Commentationes Mathematicae Universitatis Carolinae, vol. 39 (1998), no. 2, pp. 281–301.Google Scholar
[20] Shelah, Saharon and Väisänen, Pauli, The number of L∞k-equivalent nonisomorphic models for k weakly compact, Fundamenta Mathematicae, vol. 174 (2002), no. 2, pp. 97–126.Google Scholar
[21] Todorčević, Stevo and Väänänen, Jouko, Trees and Ehrenfeucht-Fraïssé games, Annals of Pure and Applied Logic, vol. 100 (1999), no. 1–3, pp. 69–97.Google Scholar
[22] Väänänen, Jouko, A Cantor-Bendixson theorem for the space , Polska Akademia Nauk. Fundamenta Mathematical vol. 137 (1991), no. 3, pp. 187–199.Google Scholar
[23] Väänänen, Jouko, Games and trees in infinitary logic: A survey, Quantifiers (Krynicki, M., Mostowski, M., and Szczerba, L., editors), Kluwer Academic Publishers, 1995, pp. 105–138.Google Scholar