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Todorchevich S., Farah I. — Some Aplications of the method of forcing
Todorchevich S., Farah I. — Some Aplications of the method of forcing

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Название: Some Aplications of the method of forcing

Авторы: Todorchevich S., Farah I.

Аннотация:

During the Fall Semester of 1991 Stevo Todorchevich (Todorcevic) gave a course on applications of the method of forcing at the Mathematical Institute in Belgrade. This text contains material presented in the course, as well as some additional closely related results included for completeness. The method of forcing, i.e. the method of adding a generic object to a given structure, is frequently used to get independent results, that is, results showing that certain statements cannot be proved (or disproved) in ZFC or some other similar theory. The main purpose of these notes is to present to a general mathematical audience a number of applications of the method of forcing to other branches of mathematics, such as general topology and measure theory. Most of the presented results do not require any additional axioms of set theory, but use standard set-theoretical and forcing constructions, such as Suslin tree, generic models^ Cohen and random reals, etc. Among topics included are Borel Equivalence Relations, Halpern-Laiichli Theorem, the Open Coloring Axiom and the Proper Forcing Axiom. For more ambitious readers there are a few exercises scattered throughout the text, and a list of yet unsolved problems is included.


Язык: en

Рубрика: Математика/

Серия: Сделано в холле

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1995

Количество страниц: 148

Добавлена в каталог: 28.10.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$MA(\kappa)$      65
$\Delta$-system      5 132
$\epsilon$-ball around h      87
$\mathcal{A}$-operation      32 128
$\mathcal{P}$-name      9
$\pi w$-homogeneous      68
$\pi$-base      66 135
$\pi$-base, point-countable      66
$\pi$-weight      66 135
Absolute class      125
Algebra of Baire sets      48
Almost disjoint subsets of $\omega$      73
Antichain (in a poset)      4
Antichain in a tree      17
Arrow-space      93 135
Automatic continuity problem      100
Axiom, Comprehension Scheme      112
Axiom, Extensionality      111
Axiom, forcing axiom      2
Axiom, forcing axioms      63
Axiom, large cardinal axiom      122
Axiom, Martin’s Axiom (MA)      65
Axiom, of choice      114
Axiom, of Foundation      118
Axiom, of infinity      116
Axiom, Open Coloring Axiom (OCA)      79
Axiom, Pairing      111
Axiom, power set      112
Axiom, Proper Forcing Axiom (PFA)      97
Axiom, Replacement Scheme      112
Axiom, union      111
Baire property      128
Baire space      18
Ball of partial functions      87
Base for a class of topological spaces      93
BP Boolean Prime Ideal      41
Branch (of a tree)      17
Caliber (of a topological space)      65
Cantor cube      1
Cardinal      115
Cardinal, limit      116
Cardinal, preserved by $\mathcal{P}$      48
Cardinal, regular      118
Cardinal, singular      118
Cardinal, strongly inaccessible      118
Cardinal, successor      116
Cardinal, weakly inaccessible      118
Cardinality      115
Cellularity, of a filter      107
Cellularity, of a topological space      21
Cellularity, of an ideal      103
Centered (family of sets)      65
Closed and unbounded subset of an ordinal      131
Closed subset of an ordinal      131
Code, for a $G_{\delta}$-set      11
Code, for an open set      11
Codes for Borel sets      130
Codes for Borel sets, $\Pi_1$      130
Codes for Borel sets, $\Sigma_1$      130
Cofinal      73
Cofinality      118
Cohen real      1
Cohen real, over M      10
Coinitial      73
Comparable      113
Complete accumulation point      137
Composant      107
Continuous family of sets      114
Continuum      66 107
Continuum Hypothesis (or CH)      116
Continuum, decomposable      107
Continuum, linearly ordered      66
Control Measure Problem      110
Correct for some formula      121
Cumulative hierarchy      118
Definition by $\rho$-recursion      114
Dense below p      1
Diadic rationals      133
Diagonalization of the sequence of sets      55
Direct product      112
Discrete subspace      135
Dominating subset of $\omega^{\omega}$      103
Elementary submodel      124
Equipotent sets      115
Exponential topology      136
Extension of a condition      1
Family of functions, trivial      85
Fills the p re-gap      73
Filter      1
Filter Dichotomy      104
Filter, $\mathcal{D}$-generic      1
Filter, Frechet      104
Filter, generic      1
Filter, M-generic      9
Forces      10
Formula, $\Delta_0$      125
Formula, $\Delta_1$      126
Formula, absolute      123
Formula, absolute for X      123
Formula, atomic      121
Formula, function-defining      112
Free sequence      136
Free sequence, of regular pairs      67
Function, Baire      48 128
Function, Borel      127
Function, choice      114
Function, order-reversing      33
Function, Skolem      124
Fusion (of the family of sets)      54
Fusion sequence      54
Gap      73 (see also “Pre-gap”)
Gap, Borel      86
Gap, Hausdorff      74
Gap, in $[\omega]^{\omega}$      73
Gap, in $\omega^{\omega}$      74
Gap, special      75
Gap, type      73
Generalized Continuum Hypothesis (GCH)      116
Group wise dense subset of $[\omega]^{\omega}$      103
Hausdorff metric      137
Height (of a node in a tree)      17
Immediate successor in a tree      41
Incomparable      1
Incompatible      1
Induction (on the ordinals)      114
Inner regularity (of a measure)      69
Interpretation (of a $\mathcal{P}$-name)      9
Interval algebra      90
Interval determined by p      1
Invariant subspace problem      110
Isomorphic elements of $C_{\theta}$      50
Isomorphism type of p      50
Large enough part of ZFC      124
Lemma, $\Delta$-system      5 132
Lemma, Fusion Lemma      54
Lemma, Pressing-Down Lemma      131
Lemma, Truth Lemma of forcing extensions      10
Level (of a tree)      17
lexicographical ordering      3
Lexicographically least infinite branch      33
Measure, $\sigma$-finite      69
Measure, counting      128
Measure, Haar      128
Measure, Lebesgue      128
Measure, product      128
Measure, regular Radon      69
Model      121
Model, generic      9
Model, ground      9
Models (a relation)      121
Mokobodzki ideal      13
NCF (Near Coherence of Filters)      107
Network      93 135
OCA(X)      79
OCA*      80
Operator, compact      107
Order type      42 113
Ordering, linear      113
Ordering, partial      113
Ordering, total      113
Ordinal      113
Ordinal, limit      113
Ordinal, successor      113
Outer regularity (of a measure)      69
P-ideal      85
Partition, open      79
Partition, Sierpiriski’s      44
Partition, Suslin (or ccc)      63
Pattern, of elements of ${0, 1}^{<\omega}$      46
Pattern, of elements of ${0, 1}^{\omega}$      58
Perfect Set Property.(PSP)      80
Perfect-set Forcing      54
Polish space      127
POSET      1 113
Poset, $\sigma$-centered      99
Poset, $\sigma$-closed      97
Poset, ccc      5
Poset, of partial functions      85
Poset, proper      97 (see also “Baumgartner 1984” “Shelah
Pre-gap      73
Pre-gap, cofinal      74
Pre-gap, equivalent      74
Pre-gap, fillable (or split table)      73
Pre-gap, Suslin pre-gap      76
Precaliber (of a topological space)      65
Prenex normal form (PNF)      123
Product of posets      1
Projection of a tree      32
Proof by $\rho$-induction      114
Proper class      111
Property of Baire      19
Quasi-ordering      42
Random real      2
Random real, over M      12
Rank      119
Reals      127
Recursion (on ordinals etc).      114
Reducible coloring, regular pair      67
Relation, antisymmetric      113
Relation, Borel (analytic, coanalytic, etc).      31
Relation, Borel (analytic, coanalytic, etc)., functional      112
Relation, Borel (analytic, coanalytic, etc)., reflexive      113
Relation, Borel (analytic, coanalytic, etc)., set-like      113
Relation, Borel (analytic, coanalytic, etc)., transitive      113
Relativization (of the formula)      121
Restriction      112
Root (of a $\Delta$-system)      5
RVM      21
Sacks real      54
Separated      135
Separates      98
Set, $F_{\sigma}$, $G_{\delta}$, $\Pi^0_{\alpha}$, $\Sigma^0_{\alpha}$      127
Set, $\mathfrak{N}_1$-dense      99
Set, $\mu$-supporting      69
Set, $\Sigma_2^1$      130
Set, 0-homogeneous      4
Set, analytic      127
Set, Borel      127
Set, coanalytic      32 127
Set, comeager      13 128
Set, countable      116
Set, dense      1
Set, empty      112
Set, extension al      120
Set, free for F      71
Set, hereditarily countable      121
Set, hereditarily finite      121
Set, homogeneous      42
Set, homogeneous for f      41
Set, infinite      116
Set, meager      2 128
Set, n-dense      49
Set, nowhere dense      128
Set, null      2
Set, of conditions      1
Set, of first category      2
Set, open (in a poset)      9
Set, partially ordered      113
Set, Skolem hull      124
Set, Sorgenfry line      93 135
Set, stationary      131
Set, stationary, transitive      113
Splits the pre-gap      73
Splitting node in a tree      41
Submeasure      110
Submeasure, exhaustive      110
Subtree      41
Subtree, of $\omega^{<\omega}\otimes\omega^{<\omega}$      32
Support of a Baire set      48
Suslin Hypothesis (SH)      17
Suslin property      5
Terminal node of a tree      33
Theorem, Baire category      128
Theorem, Bernstein — Robinson      110
Theorem, Cantor      115
Theorem, Cantor — Bernstein      115
Theorem, Compactness Theorem of Predicate Calculus      124
Theorem, Dense Set Version of $HL_d$      49
Theorem, Dilworth’s      38
Theorem, Godel’s Incompleteness      121
Theorem, Helly’s      102
Theorem, HL (Halpern — Laiichli)      41
Theorem, HL, finite version      49
Theorem, Maharam’s      25
Theorem, Mostowski’s Collapsing      120
Theorem, Ramsey’s      131
Theorem, Reflection      123
Theorem, Shoenfield’s Absoluteness      35
Theorem, Ulam’s      28
Theorem, Zermelo      115
Tightness      66 136
Topological space, $T_0$      135
Topological space, $T_1$      135
Topological space, Baire      128
Topological space, ccc      21
Topological space, cometrizable      94
Topological space, countably compact      136
Topological space, countably tight      66 136
Topological space, exponential      94
Topological space, Hausdorff (or $T_2$)      135
Topological space, regular (or $T_3$)      135
Topological space, second countable      79
Topological space, Suslin      21
Transitive closure      119
TREE      17
Tree representation, of analytic sets      32
Tree representation, of coanalytic sets      32
Tree, $\omega_1$-tree      17
Tree, Aronszajn      17
Tree, perfect      41
Tree, Suslin      17
Tree, well-founded      32
Ultrafilter, nonprincipal      52
Ultrafilter, Ramsey      55
Ultrafilter, selective      55
Vertical sections      13
Vietoris topology      136
Weakly separated      94
Weight      135
Well-founded      113
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