This research level monograph reflects the current state of the field and provides a reference for graduate students entering the field as well as for established researchers.
Author(s): Tomek Bartoszynski, Haim Judah
Publisher: A K Peters/CRC Press
Year: 1995
Language: English
Pages: 560
Cover......Page cover.djvu
Front Matter......Page _004.djvu
Contents......Page _005.djvu
1.1.A Set theory......Page 001.djvu
1.1.C Stationary sets......Page 002.djvu
1.1.D Trees......Page 003.djvu
1.1.E Large cardinals......Page 004.djvu
1.2.A Borel sets......Page 005.djvu
1.2.B Group structure......Page 007.djvu
1.2.C Projective sets......Page 009.djvu
1.3 Ideals and Cardinal Invariants......Page 011.djvu
1.3.A Bounding and dominating numbers......Page 012.djvu
1.3.B The ideal of countable subsets of K......Page 014.djvu
1.3.C The ideals of meager and measure zero sets......Page 016.djvu
1.4 Forcing......Page 018.djvu
1.4.A Proper forcing......Page 023.djvu
1.4.B The Martin Axiom......Page 025.djvu
1.4.C Composition of forcing......Page 026.djvu
1.5 Iterated Forcing......Page 027.djvu
1.5.A Quotient forcing......Page 029.djvu
1.5.B Preservation of ccc......Page 033.djvu
2.1 Null and Meager Sets......Page 035.djvu
2.2 Additivity of Category......Page 038.djvu
2.2.A Characterizations of add(M) and cof(M)......Page 041.djvu
2.3 Additivity of Measure......Page 046.djvu
2.3.A Combinatorial characterizations of add (N) and cof (N)......Page 050.djvu
2.4 Covering for Category......Page 053.djvu
2.4.A Characterizations of cov (M)......Page 054.djvu
2.5 Covering for Measure......Page 060.djvu
2.5.A Small sets......Page 061.djvu
2.5.B Small versus null sets......Page 063.djvu
2.5.C Combinatorial characterization of cov(N)......Page 067.djvu
2.5.D Characterizations of non(N)......Page 071.djvu
2.6.A Combinatorics of the ideal E......Page 073.djvu
2.6.B Cardinal invariants of the ideal E......Page 075.djvu
2.6.C Closed measure zero sets and Cohen reals......Page 080.djvu
2.7 Transitive Coefficients......Page 086.djvu
2.7.A Transitive bases for measure and category......Page 087.djvu
2.7.B Transitive additivity......Page 090.djvu
2.7.C Null- and meager-additive sets......Page 095.djvu
3.2 Random Reals......Page 099.djvu
3.2.A Cardinal invariants for N versus properties of B......Page 107.djvu
3.2.B Subalgebras of the random real algebra......Page 110.djvu
3.2.D Random real algebra and the Martin Axiom......Page 113.djvu
3.2.D Random real algebra and the Martin Axiom......Page 121.djvu
3.2.E Cardinal invariants and random reals......Page 125.djvu
3.2.F Many random reals without dominating reals......Page 132.djvu
3.3 Cohen Reals......Page 138.djvu
3.3.A Cohen reals and the Martin Axiom......Page 139.djvu
3.3.B Cardinal invariants and Cohen reals......Page 142.djvu
3.4 Amoeba Forcing......Page 147.djvu
3.4.A Cardinal invariants and Amoeba reals......Page 148.djvu
3.4.B Amoeba ideal......Page 149.djvu
3.4.C Amoeba ideal and meager sets......Page 157.djvu
3.5 Dominating Reals......Page 160.djvu
3.5.A Rank function and the cardinal invariants......Page 162.djvu
3.5.B Dominating ideal......Page 165.djvu
3.6.A Basic facts about Suslin forcing......Page 168.djvu
3.6.B Suslin forcing and cardinal invariants......Page 175.djvu
3.6.C Examples of Suslin forcings......Page 177.djvu
3.6.D Suslin forcing and Cohen reals......Page 190.djvu
3.7 Suslin Ideals......Page 193.djvu
4.1.A Filters without the Baire property......Page 205.djvu
4.1.B Nonmeasurable filters......Page 207.djvu
4.2.A A model where all nonmeasurable filters do not have the Baire property......Page 210.djvu
4.2.B Bounded nonmeasurable filters......Page 213.djvu
4.3.A Intersection of unbounded filters......Page 216.djvu
4.3.B Intersection of nonmeasurable filters......Page 217.djvu
4.3.C A model where the intersection of < 2^Aleph_0 No ultrafilters has measure zero......Page 221.djvu
4.4.A Characterizations of p-points......Page 224.djvu
4.4.B A model where there are no p-points......Page 226.djvu
4.4.C Nonmeager p-filters......Page 230.djvu
4.4.D A model where every unbounded p-filter is generated by > d elements......Page 231.djvu
4.5.A Characterizations of Ramsey filters......Page 235.djvu
4.5.B Ramsey filters and Cohen reals......Page 237.djvu
4.6 Rapid Filters......Page 239.djvu
4.6.A A model where there are no rapid filters......Page 242.djvu
4.6.B The Raisonnier filter......Page 245.djvu
5.1.A Ideal associated with cov(M)......Page 251.djvu
5.1.B Cofinality of cov(M)......Page 255.djvu
5.1.C Cofinality of cov(N)......Page 261.djvu
5.2 Consistency Results Involving Singular Cardinals......Page 263.djvu
6.1.A Preservation of properness......Page 271.djvu
6.1. B First Preservation Theorem......Page 273.djvu
6.1.C Second Preservation Theorem......Page 282.djvu
6.2 Preservation of P- Points......Page 285.djvu
6.3 Applications - Countable Support Iteration......Page 290.djvu
6.3.A Preservation of omega^omega-bounding and "no dominating reals are added"......Page 291.djvu
6.3.B Preservation of outer measure and "no random reals are added"......Page 292.djvu
6.3.C Preservation of non meager sets and preservation of bases of the ideal of meager sets......Page 295.djvu
6.3.D Preservation of the property "\bigcup(M\capV)\notinM"......Page 297.djvu
6.3.E Preservation of the Laver property and the property "old reals are not null additive"......Page 299.djvu
6.3.F Preservation of bases of the ideal of measure zero sets and the property "\bigcup(N\capV)\notinN"......Page 302.djvu
6.4 Finite Support Iteration......Page 305.djvu
6.5.A Preserving unbounded families......Page 313.djvu
6.5.B Preserving the property "\bigcup(N\capV)\notinN"......Page 317.djvu
6.5.C Preservation of nonmeager sets......Page 320.djvu
6.5.D Preservation of the property "no random reals are added"......Page 321.djvu
6.5.E Preserving the property "\bigcup(M\capV)\notinM"......Page 323.djvu
7.1 Basic Definitions and Notation......Page 325.djvu
7.2.A Property L_f......Page 327.djvu
7.2.B Property P_f......Page 328.djvu
7.2.C Property B......Page 330.djvu
7.2.D Property B_f,h......Page 331.djvu
7.2.E Preservation of not adding Cohen reals......Page 334.djvu
7.3.A Forcings associated with add(N), cof(N), and 2^aleph_0......Page 338.djvu
7.3.B Forcing associated with non(M)......Page 339.djvu
7.3.C Forcing associated with non(N)......Page 348.djvu
7.3.D Forcing associated with b......Page 352.djvu
7.3.E Forcing associated with d......Page 359.djvu
7.4.A The Mathias forcing M......Page 363.djvu
7.4.B Eventually different forcing E......Page 366.djvu
7.4.C Infinitely equal forcing EE......Page 367.djvu
7.4.D Blass-Shelah forcing Q......Page 370.djvu
7.5 Models Satisfying b = [] and d=[x]......Page 380.djvu
7.6 Models Satisfying b =[x] or d=[]......Page 387.djvu
8 Strong Measure Zero and Strongly Meager Sets......Page 398.djvu
8.1 Characterizations of Strong Measure Zero Sets......Page 399.djvu
8.1.A Strong measure zero in metric spaces......Page 400.djvu
8.1.B Strong measure zero sets and translations......Page 405.djvu
8.2.A Basic constructions in ZFC......Page 410.djvu
8.2.B Models with no strong measure zero sets of size 2^Aleph_0......Page 414.djvu
8.3 The Borel Conjecture......Page 417.djvu
8.3.A The Borel Conjecture and 2^Aleph_0 = Aleph_1......Page 418.djvu
8.3.B The Borel conjecture and 2^Aleph_0 > Aleph_1......Page 421.djvu
8.4 Additivity of the Ideal SN......Page 420.djvu
8.4.A A model where cov(SN) < add(M) ......Page 431.djvu
8.4.B Increasing add(SN) by forcing......Page 434.djvu
8.5 Strongly Meager Sets......Page 437.djvu
8.5.A A characterization of strongly meager sets ......Page 439.djvu
8.5.B The Dual Borel Conjecture......Page 447.djvu
9 .1 Sigma^1_1 Sets of Reals......Page 449.djvu
9.2.A Characterization of Delta^1_2(L) and Delta^1_2(B)......Page 452.djvu
9.2.B Sigma^1_n-P-absoluteness ......Page 456.djvu
9.3 Sigma^1_2 Sets of Reals......Page 457.djvu
9.3.A Sigma^1_2 filters on omega......Page 461.djvu
9.4 Delta^1_3 Sets of Reals......Page 462.djvu
9.4.A A model where all Delta^1_3 sets are measurable and have the Baire Property......Page 464.djvu
9.4.B MA + Delta^1_3(B)......Page 467.djvu
9.4.C Delta^1_3(L) -/-> Delta^1_3(B)......Page 471.djvu
9.5 Sigma^_3 Sets of Reals......Page 474.djvu
9.6 Sigma^1_3 Sets of Reals......Page 478.djvu
9.6.A Collapsing algebra......Page 479.djvu
9.7.A MA_Aleph_0 (Suslin) and Sigma^1_n(L)......Page 484.djvu
9.7.B MA_Aleph_0 (sigma-linked) and Sigma^1_n(L)......Page
9.7.C MA and Sigma^1_n(L)......Page 491.djvu
9.8 The Baire Property without Inaccessible Cardinals......Page 495.djvu
9.8.A Amalgamation......Page 497.djvu
9.8.B Sweet forcings......Page 501.djvu
9.8.C A model where all projective sets have the Baire property and 2^Aleph_0 is large ......Page 508.djvu
Bibligraphy ......Page 517.djvu
List of Symbols......Page 532.djvu
Index......Page 535.djvu
