The Tree Property at ω₂ and Bounded Forcing Axioms

• Autorzy: Friedman S.-D. , Torres-Pérez V.

Identyfikatory

DOI

10.4064/ba8038-1-2016

• Języki publikacji: EN

Abstrakty

EN

We prove that the Tree Property at ω₂ together with BPFA is equiconsistent with the existence of a weakly compact reflecting cardinal, and if BPFA is replaced by BPFA(ω₁) then it is equiconsistent with the existence of just a weakly compact cardinal. Similarly, we show that the Special Tree Property for ω₂ together with BPFA is equiconsistent with the existence of a reflecting Mahlo cardinal, and if BPFA is replaced by BPFA(ω₁) then it is equiconsistent with the existence of just a Mahlo cardinal.

Słowa kluczowe

EN

tree property; BPFA; Aronszajn trees; special Aronszajn trees

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2015
• Tom: Vol. 63, no. 3
• Strony: 207--216
• Opis fizyczny: Bibliogr. 16 poz.

Twórcy

autor

Friedman S.-D.

• Kurt Gödel Research Center, Universität Wien, Währinger Straße 25, A-1090 Wien, Austria

autor

Torres-Pérez V.

• Institut für Diskrete Mathematik und Geometrie, TU Wien, Wiedner Haupstraße 8/104, 1040 Wien, Austria

Bibliografia

• [1] J. Bagaria, Bounded forcing axioms as principles of generic absoluteness, Arch. Math. Logic 39 (2000), 393–401.
• [2] J. E. Baumgartner, Applications of the proper forcing axiom, in: Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, 913–959.
• [3] J. E. Baumgartner and R. Laver, Iterated perfect-set forcing, Ann. Math. Logic 17 (1979), 271–288.
• [4] J. Cummings, M. Foreman, and M. Magidor, Scales, squares and reflection, J. Math. Logic 1 (2001), 35–98.
• [5] P. Erdős and A. Tarski, On some problems involving inaccessible cardinals, in: Essays on the Foundations of Mathematics, Magnes Press, Hebrew Univ., Jerusalem, 1961, 50–82.
• [6] M. Goldstern and S. Shelah, The bounded proper forcing axiom, J. Symbolic Logic 60 (1995), 58–73.
• [7] W. P. Hanf and D. Scott, Classifying inaccessible cardinals, Notices Amer. Math. Soc. 8 (1961), 445 (abstract).
• [8] T. Jech, Set Theory, Springer Monogr. Math., Springer, Berlin, 2003.
• [9] R. B. Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229–308; Erratum, ibid. 4 (1972), 443.
• [10] K. Kunen, Set Theory, Stud. Logic (London) 34, College Publ., London, 2011.
• [11] D. Kurepa, Ensembles ordonnés et ramifiés, Publ. Math. Univ. Belgrade 4 (1935), 1–38.
• [12] W. Mitchell, Aronszajn trees and the independence of the transfer property, Ann. Math. Logic 5 (1972/73), 21–46.
• [13] J. Moore, Proper forcing, cardinal arithmetic, and uncountable linear orders, Bull. Symbolic Logic 11 (2005), 51–60.
• [14] H. Sakai and B. Veličkovic, Stationary reflection principles and two cardinal tree properties, J. Inst. Math. Jussieu 14 (2015), 69–85.
• [15] S. Shelah, Proper Forcing, Lecture Notes in Math. 940, Springer, Berlin, 1982.
• [16] W. Z. Sun, Stationary cardinals, Arch. Math. Logic 32 (1993), 429–442.