Skip to main content

-definability at uncountable regular cardinals

  • Philipp Lücke (a1)

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

Given an uncountable regular cardinal κ with κ = κ <κ and an arbitrary subset A of κ κ, we show that there is a <κ-closed forcing ℙ that satisfies the κ +-chain condition and forces A to be a -subset of κ κ 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 κ κ whose graph is a -subset of κ κ × κ κ.

(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 κ κ) 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.

Hide All
[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. 115.
[2] Asperó, David and Friedman, Sy-David, Definable wellorderings of Hω2 and CH, Preprint available online at
[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. 313.
[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. 775884.
[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. 203242.
[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. 2440.
[7] Friedman, Sy-David and Holy, Peter, Condensation and large cardinals, Fundamenta Mathematical vol. 215 (2011), no. 2, pp. 133166.
[8] Friedman, Sy-David, Hyttinen, Tapani, and Kulikov, Vadim, Generalised descriptive set theory and classification theory, Preprint available online at
[9] Fuchs, Gunter, Closed maximality principles: implications, separations and combinations, this Journal, vol. 73 (2008), no. 1, pp. 276308.
[10] Harrington, Leo, Long projective wellorderings, Annals of Pure andApplied Logic, vol. 12 (1977), no. 1, pp. 124.
[11] Hyttinen, Tapani and Rautila, Mika, The canary tree revisited, this Journal, vol. 66 (2001), no. 4, pp. 16771694.
[12] Hyttinen, Tapani and Väänänen, Jouko, On Scott and Karp trees of uncountable models, this Journal, vol. 55 (1990), no. 3, pp. 897908.
[13] Jech, Thomas J., Trees, this Journal, vol. 36 (1971), pp. 114.
[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. 84104.
[15] Kanamori, Akihiro, The higher infinite, second ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
[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. 16.
[17] Mekler, Alan and Väänänen, Jouko, Trees and -subsets of ω1 ω1 , this Journal, vol. 58 (1993), no. 3, pp. 10521070.
[18] Nadel, Mark and Stavi, Jonathan, L∞λ,-equivalence, isomorphism and potential isomorphism, Transactions of the American Mathematical Society, vol. 236 (1978), pp. 5174.
[19] Neeman, Itay and Zapletal, Jindřich, Proper forcings and absoluteness in L(R), Commentationes Mathematicae Universitatis Carolinae, vol. 39 (1998), no. 2, pp. 281301.
[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. 97126.
[21] Todorčević, Stevo and Väänänen, Jouko, Trees and Ehrenfeucht-Fraïssé games, Annals of Pure and Applied Logic, vol. 100 (1999), no. 1–3, pp. 6997.
[22] Väänänen, Jouko, A Cantor-Bendixson theorem for the space , Polska Akademia Nauk. Fundamenta Mathematical vol. 137 (1991), no. 3, pp. 187199.
[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. 105138.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed