Source: J. Symbolic Logic Volume 77, Issue 3
(2012), 1011-1046.
Let κ be an infinite cardinal. A subset of
(κκ)n is a
Σ11-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 Σ1k-, Π1k- and Δ1k-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 ℛ that satisfies the κ+-chain condition and forces A to be a Δ11-subset of κκ in every ℛ-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(𝒫(κ)).
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
Δ12-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
Σ13(κκ)-formulae
(i.e., formulae with parameters which define
Σ13-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 Σ11- and Δ11-subsets have some useful structural properties in certain ZFC-models.
References
David Asperó and Sy-David Friedman Large cardinals and locally defined well-orders of the universe, Annals of Pure and Applied Logic, vol. 157(2009), no. 1, pp. 1–15.
–––– Definable wellorderings of ${\rm{H}}_{\omega_2}$ and ${\rm{CH}}$, Preprint available online at http://www.logic.univie.ac.at/\~ sdf/papers/joint.aspero.omega-2.pdf.
Joan Bagaria and Sy D. Friedman Generic absoluteness, Proceedings of the XIth Latin American Symposium on Mathematical Logic (Mérida, 1998), vol. 108,2001, pp. 3–13.
James Cummings Iterated forcing and elementary embeddings, The handbook of set theory (M. Foreman, A. Kanamorie, and M. Magidor, editors), vol. 2, Springer, Berlin,2010, pp. 775–884.
Qi Feng, Menachem Magidor, and Hugh Woodin 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.
Sy-David Friedman 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.
Sy-David Friedman and Peter Holy Condensation and large cardinals, Fundamenta Mathematicae, vol. 215(2011), no. 2, pp. 133–166.
Sy-David Friedman, Tapani Hyttinen, and Vadim Kulikov Generalised descriptive set theory and classification theory, Preprint available online at http://www.logic.univie.ac.at/\~ sdf/papers/joint.tapani.vadim.pdf.
Gunter Fuchs Closed maximality principles: implications, separations and combinations, Journal of Symbolic Logic, vol. 73(2008), no. 1, pp. 276–308.
Leo Harrington Long projective wellorderings, Annals of Pure and Applied Logic, vol. 12(1977), no. 1, pp. 1–24.
Mathematical Reviews (MathSciNet):
MR465866
Tapani Hyttinen and Mika Rautila The canary tree revisited, Journal of Symbolic Logic, vol. 66(2001), no. 4, pp. 1677–1694.
Tapani Hyttinen and Jouko Väänänen On Scott and Karp trees of uncountable models, Journal of Symbolic Logic, vol. 55(1990), no. 3, pp. 897–908.
Thomas J. Jech Trees, Journal of Symbolic Logic, vol. 36(1971), pp. 1–14.
Mathematical Reviews (MathSciNet):
MR284331
R. B. Jensen and R. M. Solovay Some applications of almost disjoint sets, Mathematical Logic and Foundations of Set Theory, North-Holland, Amsterdam,1970, pp. 84–104.
Mathematical Reviews (MathSciNet):
MR289291
Akihiro Kanamori The higher infinite, second ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin,2003.
Richard Laver 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.
Alan Mekler and Jouko Väänänen Trees and $\Pi^1_1$-subsets of $^{\omega_1}\omega_1$, Journal of Symbolic Logic, vol. 58(1993), no. 3, pp. 1052–1070.
Mark Nadel and Jonathan Stavi $L_{\infty }{}_{\lambda }$-equivalence, isomorphism and potential isomorphism, Transactions of the American Mathematical Society, vol. 236(1978), pp. 51–74.
Mathematical Reviews (MathSciNet):
MR462942
Itay Neeman and Jindřich Zapletal Proper forcings and absoluteness in $L(\mathbf{R})$, Commentationes Mathematicae Universitatis Carolinae, vol. 39(1998), no. 2, pp. 281–301.
Saharon Shelah and Pauli Väisänen The number of $L_{\infty\kappa}$-equivalent nonisomorphic models for $\kappa$ weakly compact, Fundamenta Mathematicae, vol. 174(2002), no. 2, pp. 97–126.
Stevo Todorčević and Jouko Väänänen Trees and Ehrenfeucht-Fraï ssé games, Annals of Pure and Applied Logic, vol. 100(1999), no. 1-3, pp. 69–97.
Jouko Väänänen A Cantor-Bendixson theorem for the space $\omega^{\omega_1}_1$, Polska Akademia Nauk. Fundamenta Mathematicae, vol. 137(1991), no. 3, pp. 187–199.
–––– Games and trees in infinitary logic: A survey, Quantifiers (M. Krynicki, M. Mostowski, and L. Szczerba, editors), Kluwer Academic Publishers,1995, pp. 105–138.
