LEADER 04937oam 2200589 450 001 9910790973803321 005 20190911112729.0 010 $a981-4571-58-X 035 $a(OCoLC)869457463 035 $a(MiFhGG)GVRL8QYR 035 $a(EXLCZ)992550000001191485 100 $a20131025h20142014 uy 0 101 0 $aeng 135 $aurun|---uuuua 181 $ctxt 182 $cc 183 $acr 200 10$aNotes on forcing axioms /$fStevo Todorcevic, University of Toronto, Canada ; editors, Chitat Chong, Qi Feng, Yue Yang, National University of Singapore, Singapore, Theodore A. Slaman, W. Hugh Woodin, University of California, Berkeley, USA 210 1$aNew Jersey :$cWorld Scientific,$d[2014] 210 4$d?2014 215 $a1 online resource (xiii, 219 pages) $cillustrations 225 1 $aLecture notes series (Institute for Mathematical Sciences, National University of Singapore),$x1793-0758 ;$vvolume 26 300 $aDescription based upon print version of record. 311 $a981-4571-57-1 311 $a1-306-39657-3 320 $aIncludes bibliographical references. 327 $aContents; Foreword by Series Editors; Foreword by Volume Editors; Preface; 1 Baire Category Theorem and the Baire Category Numbers; 1.1 The Baire category method - a classical example; 1.2 Baire category numbers; 1.3 P-clubs; 1.4 Baire category numbers of posets; 1.5 Proper and semi-proper posets; 2 Coding Sets by the Real Numbers; 2.1 Almost-disjoint coding; 2.2 Coding families of unordered pairs of ordinals; 2.3 Coding sets of ordered pairs; 2.4 Strong coding; 2.5 Solovay's lemma and its corollaries; 3 Consequences in Descriptive Set Theory; 3.1 Borel isomorphisms between Polish spaces 327 $a3.2 Analytic and co-analytic sets 3.3 Analytic and co-analytic sets under p > ?1; 4 Consequences in Measure Theory; 4.1 Measure spaces; 4.2 More on measure spaces; 5 Variations on the Souslin Hypothesis; 5.1 The countable chain condition; 5.2 The Souslin Hypothesis; 5.3 A selective ultrafilter from m > ?1; 5.4 The countable chain condition versus the separability; 6 The S-spaces and the L-spaces; 6.1 Hereditarily separable and hereditarily Lindelof spaces; 6.2 Countable tightness and the S- and L-space problems; 7 The Side-condition Method; 7.1 Elementary submodels as side conditions 327 $a7.2 Open graph axiom 8 Ideal Dichotomies; 8.1 Small ideal dichotomy; 8.2 Sparse set-mapping principle; 8.3 P-ideal dichotomy; 9 Coherent and Lipschitz Trees; 9.1 The Lipschitz condition; 9.2 Filters and trees; 9.3 Model rejecting a finite set of nodes; 9.4 Coloring axiom for coherent trees; 10 Applications to the S-space Problem and the von Neumann Problem; 10.1 The S-space problem and its relatives; 10.2 The P-ideal dichotomy and a problem of von Neumann; 11 Biorthogonal Systems; 11.1 The quotient problem; 11.2 A topological property of the dual ball; 11.3 A problem of Rolewicz 327 $a16 Cardinal Arithmetic and mm 16.1 mm and the continuum; 16.2 mm and cardinal arithmetic above the continuum; 17 Reflection Principles; 17.1 Strong reflection of stationary sets; 17.2 Weak reflection of stationary sets; 17.3 Open stationary set-mapping reflection; Appendix A Basic Notions; A.1 Set theoretic notions; A.2 ?-systems and free sets; A.3 Topological notions; A.4 Boolean algebras; Appendix B Preserving Stationary Sets; B.1 Stationary sets; B.2 Partial orders, Boolean algebras and topological spaces; B.3 A topological translation of stationary set preserving 327 $aAppendix C Historical and Other Comments 330 $aIn the mathematical practice, the Baire category method is a tool for establishing the existence of a rich array of generic structures. However, in mathematics, the Baire category method is also behind a number of fundamental results such as the Open Mapping Theorem or the Banach-Steinhaus Boundedness Principle. This volume brings the Baire category method to another level of sophistication via the internal version of the set-theoretic forcing technique. It is the first systematic account of applications of the higher forcing axioms with the stress on the technique of building forcing notions 410 0$aLecture notes series (National University of Singapore. Institute for Mathematical Sciences) ;$vv. 26. 606 $aForcing (Model theory) 606 $aAxioms 606 $aBaire classes 615 0$aForcing (Model theory) 615 0$aAxioms. 615 0$aBaire classes. 676 $a511.3 700 $aTodorcevic$b Stevo$061532 702 $aChong$b C.-T$g(Chi-Tat),$f1949- 702 $aFeng$b Qi$f1955- 702 $aYang$b Yue$f1964- 702 $aSlaman$b T. A$g(Theodore Allen),$f1954- 702 $aWoodin$b W. H$g(W. Hugh), 801 0$bMiFhGG 801 1$bMiFhGG 906 $aBOOK 912 $a9910790973803321 996 $aNotes on forcing axioms$93871606 997 $aUNINA