Abstract
We build a model of the P-ideal ichotomy (PID) and Martin’s axiom for ω1 (\({\rm MA}_{{\omega}_1}\)) in which there is a 2-entangled set of reals. In particular, it follows that the Open Graph Axiom or Baumgartner’s axiom for ω1-dense sets are not consequences of \({\rm PID} + {\rm MA}_{{\omega}_1}\). We review Neeman’s iteration method using two type side conditions and provide an alternative proof for the preservation of properness.
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References
U. Abraham, Proper forcing, in Handbook of Set Theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 333–394.
U. Abraham and S. Todorčević, Martin’s axiom and first-countable S- and L-spaces, in Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, pp. 327–346.
U. Abraham and S. Todorčević, Partition properties of ω1 compatible with CH, Fundamenta Mathematica 152 (1997), 165–181.
D. Asperó and M. A. Mota, Forcing consequences of PFA together with the continuum large, Transactions of the American Mathematical Society 367 (2015), 6103–6129.
D. Asperó and M. A. Mota, Measuring club-sequences together with the continuum large, Journal of Symbolic Logic 82 (2017), 1066–1079.
U. Avraham and S. Shelah, Martin’s axiom does not imply that every two ℵ1-dense sets of reals are isomorphic, Israel Journal of Mathematics 38 (1981), 161–176.
B. Balcar and T. Jech, Weak distributivity, a problem of von Neumann and the mystery of measurability, Bulletin of Symbolic Logic 12 (2006), 241–266.
J. E. Baumgartner, All ℵ1-dense sets of reals can be isomorphic, Fundamenta Mathematica 79 (1973), 101–106.
J. E. Baumgartner, Applications of the proper forcing axiom, in Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, pp. 913–959.
M. Bekkali, Topics in Set Theory, Lecture Notes in Mathematics, Vol. 1476, Springer, Berlin, 1991.
A. Blass, Combinatorial cardinal characteristics of the continuum, in Handbook of Set Theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 395–489.
P. Borodulin-Nadzieja and D. Chodounský, Hausdorff gaps and towers in \({\cal P}(\omega)\)/Fin, Fundamenta Mathematica 229 (2015), 197–229.
D. Chodounský and J. Zapletal, Why Y-c.c, Annals of Pure and Applied Logic 166 (2015), 1123–1149.
A. Dow, Generalized side-conditions and Moore–Mrówka, Topology and its Applications 197 (2016), 75–101.
I. Farah, OCA and towers in \({\cal P}({\rm N})\)/fin, Commentationes Mathematicae Universitatis Carolinae 37 (1996), 861–866.
I. Farah, Analytic quotients: theory of liftings for quotients over analytic ideals on the integers, Memoirs of the American Mathematical Society 148 (2000).
I. Farah, All automorphisms of the Calkin algebra are inner, Annals of Mathematics 173 (2011), 619–661.
D. H. Fremlin, Consequences of Martin’s Axiom, Cambridge Tracts in Mathematics, Vol. 84, Cambridge University Press, Cambridge, 1984.
M. Gitik and M. Magidor, SPFA by finite conditions, Archive for Mathematical Logic 55 (2016), 649–661.
P. Holy, P. Lücke and A. Njegomir, Characterizing large cardinals through Neeman’s pure side condition forcing, Fundamenta Mathematica 252 (2021), 53–102.
T. Ishiu and J. T. Moore, Minimality of non-σ-scattered orders, Fundamenta Mathematica 205 (2009), 29–44.
T. Jech, Set Theory, Springer Monographs in Mathematics, Springer, Berlin, 2003.
K. Kunen, Set Theory, Studies in Logic and the Foundations of Mathematics, Vol. 102, North-Holland, Amsterdam-New York, 1980.
K. Kunen, Set Theory, Studies in Logic (London), Vol. 34, College Publications, London, 2011.
B. Kuzeljevic and S. Todorcevic, P-ideal dichotomy and a strong form of the Suslin hypothesis, Fundamenta Mathematica 251 (2020), 17–33.
H. Lamei Ramandi, A new minimal non-σ-scattered linear order, Journal of Symbolic Logic 84 (2019), 1576–1589.
H. Lamei Ramandi and J. T. Moore, There may be no minimal non-σ-scattered linear orders, Mathematical Research Letters 25 (2018), 1957–1975.
P. Larson, Showing OCA in ℙmax-style extensions, Kobe Journal of Mathematics 18 (2001), 115–126.
R. Laver, Making the supercompactness of κ indestructible under κ-directed closed forcing, Israel Journal of Mathematics 29 (1978), 385–388.
H. Mildenberger and L. Zdomskyy, L-spaces and the P-ideal dichotomy, Acta Mathematica Hungarica 125 (2009), 85–97.
T. Miyamoto and T. Yorioka, A fragment of Asperó–Mota’s finitely proper forcing axiom and entangled sets of reals, Fundamenta Mathematica 251 (2020), 35–68.
J. T. Moore, Open colorings, the continuum and the second uncountable cardinal, Proceedings of the American Mathematical Society 130 (2002), 2753–2759.
J. T. Moore, A five element basis for the uncountable linear orders, Annals of Mathematics 163 (2006), 669–688.
J. T. Moore, ω1 and −ω1 may be the only minimal uncountable linear orders, Michigan Mathematical Jouenal 55 (2007), 437–457.
J. T. Moore, The proper forcing axiom, in Proceedings of the International Congress of Mathematicians. Vol. II, Hindustan Book Agency, New Delhi, 2010, pp. 3–29.
J. T. Moore and S. Todorcevic, Baumgartner’s isomorphism problem for ℵ2-dense sub-orders of ℝ, Archive for Mathematical Logic 56 (2017), 1105–1114.
I. Neeman, Forcing with sequences of models of two types, Notre Dame Journal of Formal Logic 55 (2014), 265–298.
I. Neeman, Two applications of finite side conditions at ω2, Archive for Mathematical Logic 56 (2017), 983–1036.
D. Raghavan, P-ideal dichotomy and weak squares, Journal of Symbolic Logic 78 (2013), 157–167.
D. Raghavan and S. Todorcevic, Combinatorial dichotomies and cardinal invariants, Mathematical Research Letters 21 (2014), 379–401.
S. Shelah, Proper and Improper Forcing, Perspectives in Mathematical Logic, Springer, Berlin, 1998.
L. Soukup, Indestructible properties of S- and L-spaces, Topology and its Applications 112 (2001), 245–257.
J. Steprāns and W. S. Watson, Homeomorphisms of manifolds with prescribed behaviour on large dense sets, Bulletin of the London Mathematical Society 19 (1987), 305–310.
S. Todorčević, Directed sets and cofinal types, Transactions of the American Mathematical Society 290 (1985), 711–723.
S. Todorčević, Remarks on chain conditions in products, Compositio Mathematica 55 (1985), 295–302.
S. Todorčević, Remarks on cellularity in products, Compositio Mathematica 57 (1986), 357–372.
S. Todorčević, Partition Problems in Topology, Contemporary Mathematics, Vol. 84, American Mathematical Society, Providence, RI, 1989.
S. Todorčević, A dichotomy for P-ideals of countable sets, Fundamenta Mathematica 166 (2000), 251–267.
S. Todorcevic, A proof of Nogura’s conjecture, Proceedings of the American Mathematical Society 131 (2003), 3919–3923.
S. Todorcevic, Walks on Ordinals and Their Characteristics, Progress in Mathematics, Vol. 263, Birkhäuser, Basel, 2007.
S. Todorcevic, Combinatorial dichotomies in set theory, Bulletin of Symbolic Logic 17 (2011), 1–72.
S. Todorcevic, Notes on Forcing Axioms, Lecture Notes Series. Institute for Mathematical Sciences. National University of Singapore, Vol. 26, World Scientific, Hackensack, NJ, 2014.
S. Todorchevich and I. Farah, Some applications of the method of forcing, Yenisei Series in Pure and Applied Mathematics, Yenisei, Moscow, 1995.
B. Veličković, Applications of the open coloring axiom, in Set Theory of the Continuum (Berkeley, CA, 1989), Mathematical Sciences Research Institute Publications, Vol. 26, Springer, New York, 1992, pp. 137–154.
B. Veličković and G. Venturi, Proper forcing remastered, in Appalachian Set Theory 2006–2012, London Mathematical Society Lecture Note Series, Vol. 406, Cambridge University Press, Cambridge, 2013, pp. 331–362.
G. Venturi, Preservation of Suslin trees and side conditions, Journal of Symbolic Logic 81 (2016), 483–492.
M. Viale, A family of covering properties, Mathematical Research Letters 15 (2008), 221–238.
T. Yorioka, A note on a forcing related to the S-space problem in the extension with a coherent Suslin tree, Mathematical Logic Quarterly 61 (2015), 169–178.
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We would like to thank the referee for their comments.
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The first author was supported by a PAPIIT grant IA102222.
The second author is partially supported by grants from NSERC (455916), CNRS (IMJ-PRG-UMR7586) and SFRS (7750027-SMART).
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Guzmán, O., Todorcevic, S. The P-ideal dichotomy, Martin’s axiom and entangled sets. Isr. J. Math. (2024). https://doi.org/10.1007/s11856-024-2651-8
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DOI: https://doi.org/10.1007/s11856-024-2651-8