Separating club-guessing principles in the presence of fat forcing axioms
We separate various weak forms of Club Guessing at ω1 in the presence of 2ℵ0 large, Martin's Axiom, and related forcing axioms. We also answer a question of Abraham and Cummings concerning the consistency of the failure of a certain polychromatic Ramsey statement together with the continuum lar...
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| Published in | Annals of pure and applied logic Vol. 167; no. 3; pp. 284 - 308 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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Elsevier B.V
01.03.2016
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| Subjects | |
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| Abstract | We separate various weak forms of Club Guessing at ω1 in the presence of 2ℵ0 large, Martin's Axiom, and related forcing axioms.
We also answer a question of Abraham and Cummings concerning the consistency of the failure of a certain polychromatic Ramsey statement together with the continuum large.
All these models are generic extensions via finite support iterations with symmetric systems of structures as side conditions, possibly enhanced with ω-sequences of predicates, and in which the iterands are taken from a relatively small class of forcing notions.
We also prove that the natural forcing for adding a large symmetric system of structures (the first member in all our iterations) adds ℵ1-many reals but preserves CH. |
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| AbstractList | We separate various weak forms of Club Guessing at ω1 in the presence of 2ℵ0 large, Martin's Axiom, and related forcing axioms.
We also answer a question of Abraham and Cummings concerning the consistency of the failure of a certain polychromatic Ramsey statement together with the continuum large.
All these models are generic extensions via finite support iterations with symmetric systems of structures as side conditions, possibly enhanced with ω-sequences of predicates, and in which the iterands are taken from a relatively small class of forcing notions.
We also prove that the natural forcing for adding a large symmetric system of structures (the first member in all our iterations) adds ℵ1-many reals but preserves CH. |
| Author | Mota, Miguel Angel Asperó, David |
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| Cites_doi | 10.1016/j.apal.2008.07.006 10.1007/BF02803525 10.1007/s11856-015-1250-0 10.2178/jsl/1333566635 10.1016/j.apal.2014.12.001 10.2307/2000309 10.2178/jsl/1191333845 10.4310/MRL.1998.v5.n2.a8 10.1090/S0002-9947-2015-06205-9 10.2478/s11533-012-0037-3 |
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| Keywords | 03E50 Iterated forcing 03E57 03E35 Club-guessing principles Polychromatic Ramsey theory 03E05 Side conditions |
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| References | Galvin (br0090) 1975; 82 Shelah (br0200) 2000 Todorčević (br0220) 1989; vol. 84 Shelah (br0190) 1998 Asperó (br0030) 2014; vol. 23 Abraham, Cummings (br0010) 2012; 10 Ishiu, Larson (br0110) 2012; 77 Asperó, Mota (br0070) 2015; 210 Zapletal (br0240) 1996; vol. 806 Koszmider (br0140) 2000; 118 Kunen (br0150) 1980 P. Nyikos, Antidiamond and anti- by finite support product of Cohen forcing, unpublished note. axioms and topological applications [incomplete preliminary version], unpublished note. Baumgartner (br0080) 1984 Asperó, Mota (br0060) 2015; 367 Abraham, Cummings, Smyth (br0020) 2007; 72 Asperó, Friedman (br0050) 2009; 157 Ishiu (br0100) 2015; 166 Zapletal (br0250) 1998; 5 Jech (br0120) 2002 Todorčević (br0210) 1985; 290 Juhász (br0130) 1988; 29 T. Yorioka, A comment on Asperó–Mota iteration, preprint, 2013. H. Sakai, Preservation of D. Asperó, Adding many Baumgartner clubs, forthcoming in a special volume to the memory of J. Baumgartner. Asperó (10.1016/j.apal.2015.12.003_br0030) 2014; vol. 23 Asperó (10.1016/j.apal.2015.12.003_br0050) 2009; 157 Juhász (10.1016/j.apal.2015.12.003_br0130) 1988; 29 10.1016/j.apal.2015.12.003_br0170 Abraham (10.1016/j.apal.2015.12.003_br0010) 2012; 10 10.1016/j.apal.2015.12.003_br0180 10.1016/j.apal.2015.12.003_br0040 Koszmider (10.1016/j.apal.2015.12.003_br0140) 2000; 118 Todorčević (10.1016/j.apal.2015.12.003_br0210) 1985; 290 Todorčević (10.1016/j.apal.2015.12.003_br0220) 1989; vol. 84 Kunen (10.1016/j.apal.2015.12.003_br0150) 1980 Asperó (10.1016/j.apal.2015.12.003_br0060) 2015; 367 Zapletal (10.1016/j.apal.2015.12.003_br0240) 1996; vol. 806 Zapletal (10.1016/j.apal.2015.12.003_br0250) 1998; 5 Galvin (10.1016/j.apal.2015.12.003_br0090) 1975; 82 Ishiu (10.1016/j.apal.2015.12.003_br0100) 2015; 166 Ishiu (10.1016/j.apal.2015.12.003_br0110) 2012; 77 Shelah (10.1016/j.apal.2015.12.003_br0200) Asperó (10.1016/j.apal.2015.12.003_br0070) 2015; 210 10.1016/j.apal.2015.12.003_br0230 Jech (10.1016/j.apal.2015.12.003_br0120) 2002 Baumgartner (10.1016/j.apal.2015.12.003_br0080) 1984 Shelah (10.1016/j.apal.2015.12.003_br0190) 1998 Abraham (10.1016/j.apal.2015.12.003_br0020) 2007; 72 |
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Soc. contributor: fullname: Todorčević – volume: vol. 806 start-page: 476 year: 1996 end-page: 484 ident: br0240 article-title: Characterization of the club forcing publication-title: Papers on General Topology and Applications contributor: fullname: Zapletal – volume: vol. 23 start-page: 5 year: 2014 end-page: 27 ident: br0030 article-title: The consistency of a club-guessing failure at the successor of a regular cardinal publication-title: Infinity, Computability, and Methamathematics: Festschrift Celebrating the 60th Birthdays of Peter Koepke and Philip Welch contributor: fullname: Asperó – year: 1984 ident: br0080 article-title: Applications of the Proper Forcing Axiom publication-title: Handbook of Set Theoretic Topology contributor: fullname: Baumgartner – volume: 72 start-page: 865 year: 2007 end-page: 896 ident: br0020 article-title: Some results in polychromatic Ramsey theory publication-title: J. Symbolic Logic contributor: fullname: Smyth – year: 2002 ident: br0120 article-title: Set Theory: The Third Millenium Edition, Revised and Expanded contributor: fullname: Jech – volume: vol. 84 year: 1989 ident: br0220 article-title: Partition Problems in Topology publication-title: Contemporary Math. contributor: fullname: Todorčević – year: 1980 ident: br0150 article-title: Set Theory, an Introduction to Independence Proofs contributor: fullname: Kunen – volume: 29 start-page: 767 year: 1988 end-page: 773 ident: br0130 article-title: A weakening of ♣, with applications to topology publication-title: Comment. Math. Univ. Carolin. contributor: fullname: Juhász – volume: 367 start-page: 6103 year: 2015 end-page: 6129 ident: br0060 article-title: Forcing consequences of PFA together with the continuum large publication-title: Trans. Amer. Math. Soc. contributor: fullname: Mota – volume: 210 start-page: 193 year: 2015 end-page: 231 ident: br0070 article-title: A generalization of Martin's Axiom publication-title: Israel J. Math. contributor: fullname: Mota – volume: 10 start-page: 1004 year: 2012 end-page: 1016 ident: br0010 article-title: More results in polychromatic Ramsey theory publication-title: Cent. Eur. J. Math. contributor: fullname: Cummings – volume: 166 start-page: 583 year: 2015 end-page: 600 ident: br0100 article-title: The comparison of various club guessing principles publication-title: Ann. Pure Appl. Logic contributor: fullname: Ishiu – volume: 157 start-page: 1 year: 2009 end-page: 15 ident: br0050 article-title: Large cardinals and locally defined well-orders of the universe publication-title: Ann. Pure Appl. Logic contributor: fullname: Friedman – volume: 77 start-page: 515 year: 2012 end-page: 531 ident: br0110 article-title: Some results about publication-title: J. Symbolic Logic contributor: fullname: Larson – year: 1998 ident: br0190 article-title: Proper and Improper Forcing contributor: fullname: Shelah – year: 2000 ident: br0200 article-title: NNR revisited contributor: fullname: Shelah – volume: 5 start-page: 213 year: 1998 end-page: 226 ident: br0250 article-title: A dichotomy for forcing notions publication-title: Math. Res. Lett. contributor: fullname: Zapletal – volume: 118 start-page: 289 year: 2000 end-page: 315 ident: br0140 article-title: On strong chains of uncountable functions publication-title: Israel J. Math. contributor: fullname: Koszmider – volume: 82 start-page: 529 year: 1975 ident: br0090 article-title: Advanced problems: 6034 publication-title: Amer. Math. Monthly contributor: fullname: Galvin – volume: 157 start-page: 1 issue: 1 year: 2009 ident: 10.1016/j.apal.2015.12.003_br0050 article-title: Large cardinals and locally defined well-orders of the universe publication-title: Ann. Pure Appl. Logic doi: 10.1016/j.apal.2008.07.006 contributor: fullname: Asperó – year: 1984 ident: 10.1016/j.apal.2015.12.003_br0080 article-title: Applications of the Proper Forcing Axiom contributor: fullname: Baumgartner – volume: vol. 23 start-page: 5 year: 2014 ident: 10.1016/j.apal.2015.12.003_br0030 article-title: The consistency of a club-guessing failure at the successor of a regular cardinal contributor: fullname: Asperó – year: 2002 ident: 10.1016/j.apal.2015.12.003_br0120 contributor: fullname: Jech – volume: 118 start-page: 289 year: 2000 ident: 10.1016/j.apal.2015.12.003_br0140 article-title: On strong chains of uncountable functions publication-title: Israel J. Math. doi: 10.1007/BF02803525 contributor: fullname: Koszmider – volume: vol. 806 start-page: 476 year: 1996 ident: 10.1016/j.apal.2015.12.003_br0240 article-title: Characterization of the club forcing contributor: fullname: Zapletal – volume: 210 start-page: 193 year: 2015 ident: 10.1016/j.apal.2015.12.003_br0070 article-title: A generalization of Martin's Axiom publication-title: Israel J. Math. doi: 10.1007/s11856-015-1250-0 contributor: fullname: Asperó – volume: 77 start-page: 515 issue: 2 year: 2012 ident: 10.1016/j.apal.2015.12.003_br0110 article-title: Some results about (+) proved by iterated forcing publication-title: J. Symbolic Logic doi: 10.2178/jsl/1333566635 contributor: fullname: Ishiu – volume: vol. 84 year: 1989 ident: 10.1016/j.apal.2015.12.003_br0220 article-title: Partition Problems in Topology contributor: fullname: Todorčević – ident: 10.1016/j.apal.2015.12.003_br0180 – ident: 10.1016/j.apal.2015.12.003_br0200 contributor: fullname: Shelah – volume: 82 start-page: 529 issue: 5 year: 1975 ident: 10.1016/j.apal.2015.12.003_br0090 article-title: Advanced problems: 6034 publication-title: Amer. Math. Monthly contributor: fullname: Galvin – volume: 29 start-page: 767 issue: 4 year: 1988 ident: 10.1016/j.apal.2015.12.003_br0130 article-title: A weakening of ♣, with applications to topology publication-title: Comment. Math. Univ. Carolin. contributor: fullname: Juhász – volume: 166 start-page: 583 issue: 5 year: 2015 ident: 10.1016/j.apal.2015.12.003_br0100 article-title: The comparison of various club guessing principles publication-title: Ann. Pure Appl. Logic doi: 10.1016/j.apal.2014.12.001 contributor: fullname: Ishiu – volume: 290 start-page: 711 issue: 2 year: 1985 ident: 10.1016/j.apal.2015.12.003_br0210 article-title: Directed sets and cofinal types publication-title: Trans. Amer. Math. Soc. doi: 10.2307/2000309 contributor: fullname: Todorčević – volume: 72 start-page: 865 issue: 3 year: 2007 ident: 10.1016/j.apal.2015.12.003_br0020 article-title: Some results in polychromatic Ramsey theory publication-title: J. Symbolic Logic doi: 10.2178/jsl/1191333845 contributor: fullname: Abraham – ident: 10.1016/j.apal.2015.12.003_br0230 – ident: 10.1016/j.apal.2015.12.003_br0040 – ident: 10.1016/j.apal.2015.12.003_br0170 – volume: 5 start-page: 213 year: 1998 ident: 10.1016/j.apal.2015.12.003_br0250 article-title: A dichotomy for forcing notions publication-title: Math. Res. Lett. doi: 10.4310/MRL.1998.v5.n2.a8 contributor: fullname: Zapletal – volume: 367 start-page: 6103 issue: 9 year: 2015 ident: 10.1016/j.apal.2015.12.003_br0060 article-title: Forcing consequences of PFA together with the continuum large publication-title: Trans. Amer. Math. Soc. doi: 10.1090/S0002-9947-2015-06205-9 contributor: fullname: Asperó – year: 1980 ident: 10.1016/j.apal.2015.12.003_br0150 contributor: fullname: Kunen – year: 1998 ident: 10.1016/j.apal.2015.12.003_br0190 contributor: fullname: Shelah – volume: 10 start-page: 1004 issue: 3 year: 2012 ident: 10.1016/j.apal.2015.12.003_br0010 article-title: More results in polychromatic Ramsey theory publication-title: Cent. Eur. J. Math. doi: 10.2478/s11533-012-0037-3 contributor: fullname: Abraham |
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| Snippet | We separate various weak forms of Club Guessing at ω1 in the presence of 2ℵ0 large, Martin's Axiom, and related forcing axioms.
We also answer a question of... |
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| SubjectTerms | Club-guessing principles Iterated forcing Polychromatic Ramsey theory Side conditions |
| Title | Separating club-guessing principles in the presence of fat forcing axioms |
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