Abstract
We continue our earlier paper [20] by proving the equivalence, for regularκ>ω, of the existence of (κ, 1) morasses with built-in ♦ sequences and a strengthening, SK◊ , of the forcing principle, SK◊ of [20]. We obtain various applications of SK◊, to wit: the existence of a stationary subset of [K+]<K with sup as coding function, the existence of a counterexample to Arhangel’skii’s conjecture (κ=ℵ1) and compactness, axiomatizability and transfer properties for the Magidor-Malitz language ℒ\([Q_1^{< \omega } ,Q_2^1 ]\) (κ=ℵ1).
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References
J. P. Burgess,Consistency proofs in model theory: A contribution to Jensenlehre, Ann. Math. Logic14 (1978), 1–12.
J. P. Burgess,On a set-mapping problem of Hajnal and Máté, Acta Sci. Math.41 (1979), 283–288.
K. J. Devlin,Aspects of Constructibility, Springer Lecture Notes in Math.354, 1973.
K. J. Devlin,Hierarchies of constructible sets, Ann. Math. Logic11 (1977), 195–202.
K. J. Devlin and H. Johnsbraten,The Souslin Problem, Lecture Notes in Math.405, Springer-Verlag, 1974.
H. D. Donder,Another look at gap-1morasses, preprint.
H. D. Donder, R. B. Jensen and L. J. Stanley,Condensation Coherent Global Square Systems, Proceedings of Symposia in Pure Math., Cornell 1982, AMS, to appear.
A. Hajnal and I. Juhász, private communication.
R. B. Jensen,Morasses, handwritten notes,ca 1971.
I. Juhász,Cardinal Functions in Topology, Mathematisch Centrum, Amsterdam, 1975.
A. Kanamori,On Silver’s and related principles, inLogic Colloquium ’80 (Prague 1980) (D. van Dalen et al., eds.), North-Holland, Amsterdam, 1982, pp. 153–172.
A. Kanamori,Morass-level combinatorial principles, inPatras Logic Symposion (G. Metakides, ed.), North-Holland, Amsterdam, 1982, pp. 339–358.
S. Kusciuk, private communication.
M. Magidor and J. Malitz,Compact extensions of ℒ[Q],Ia, Ann. Math. Logic11 (1977), 217–261.
L. Manevitz and A. Miller,Lindelöf models of the reals: Solution to a problem of Sikorski, Isr. J. Math., to appear.
R. Price,S. Shelah’s answer to Arhangel’skii’s problem, preprint.
S. Shelah,Proper Forcing, Lecture Notes in Math.940, Springer-Verlag, 1982.
S. Shelah,On some problems in general topology, preprint.
S. Shelah,The existence of coding sets, preprint.
S. Shelah and L. Stanley,S-Forcing, I. A “black-box” theorem for morasses, with applications to super-Souslin trees, Isr. J. Math.43 (1982), 185–224.
R. M. Solovay,Box sequences and forcing, handwritten manuscript, March, 1977.
L. Stanley,“L-like” models of set theory: Forcing, combinatorial principles and morasses, Ph.D. dissertation, University of California, 1977.
L. Stanley,A short course on gap-1-morasses with areview of the fine structure of L, inSurveys in Set Theory (A. R. D. Mathias, ed.), London Math. Soc. Lecture Notes Series, 87, Cambridge Univ. Press. 1983, pp. 197–244.
S. Todorcevic,Trees, subtrees and order types, Ann. Math. Logic20 (1981), 233–268.
D. J. Velleman,Morasses, diamond and forcing, Ann. Math. Logic23 (1982), 199–281.
D. J. Velleman,Souslin trees constructed from morasses, inAxiomatic Set Theory (J. Baumgartner et al., eds.), Contemporary Mathematics, AMS, 1984.
D. J. Velleman,Review, J. Symb. Logic49 (1984), 663–665.
D. J. Velleman,Simplified morasses with linear limits, preprint.
W. S. Zwicker,P K(λ)Combinatorics I: Stationary coding sets rationalize the club filter, inAxiomatic Set Theory (J. Baumgartner et al., eds.), Contemporary Mathematics, AMS, 1984.
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Research partially supported by NSF Grant MCS 8301042.
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Shelah, S., Stanley, L.J. & Burgess, J.P. S-forcing IIa: Adding diamonds and more applications: Coding sets, Arhangel’skii’s problem and ℒ\([Q_1^{< \omega } ,Q_2^1 ]\) . Israel J. Math. 56, 1–65 (1986). https://doi.org/10.1007/BF02776239
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DOI: https://doi.org/10.1007/BF02776239