Descriptive Set Theory and Forcing - Arnold Miller

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Résumé :
1 What are the reals, anyway?.- I On the length of Borel hierarchies.- 2 Borel Hierarchy.- 3 Abstract Borel hierarchies.- 4 Characteristic function of a sequence.- 5 Martin's Axiom.- 6 Generic G?.- 7 ?-forcing.- 8 Boolean algebras.- 9 Borel order of a field of sets.- 10 CH and orders of separable metric spaces.- 11 Martin-Solovay Theorem.- 12 Boolean algebra of order ?1.- 13 Luzin sets.- 14 Cohen real model.- 15 The random real model.- 16 Covering number of an ideal.- II Analytic sets.- 17 Analytic sets.- 18 Constructible well-orderings.- 19 Hereditarily countable sets.- 20 Shoenfield Absoluteness.- 21 Mansfield-Solovay Theorem.- 22 Uniformity and Scales.- 23 Martin's axiom and Constructibility.- 24 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfKttLearuavP1wzZbItLDhis9wBH5garm % Wu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9 % q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir % -Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGa % aeqabaWaaeaaeaaakeaacqGHris5daqhaaWcbaGaeGOmaidabaGaeG % ymaedaaaaa!3322! $$ \sum _2^1 $$ well-orderings.- 25 Large % aeqabaWaaeaaeaaakeaacqGHpis1daqhaaWcbaGaeGOmaidabaGaeG % ymaedaaaaa!3310! \prod _2^1 $$ sets.- III Classical Separation Theorems.- 26 Souslin-Luzin Separation Theorem.- 27 Kleene Separation Theorem.- 28 %aeqabaWaaeaaeaaakeaacqGHpis1daqhaaWcbaGaeGymaedabaGaeG % ymaedaaaaa!330E! \prod _1^1 $$-Reduction.- 29 % aeqabaWaaeaaeaaakeaacqGHuoardaqhaaWcbaGaeGymaedabaGaeG % ymaedaaaaa!32E3! \Delta _1^1 $$-codes.- IV Gandy Forcing.- 30 % aeqabaWaaeaaeaaakeaacqGHpis1daqhaaWcbaGaeGymaedabaGaeG $$ equivalence relations.- 31 Borel metric spaces and lines in the plane.- 32 % aeqabaWaaeaaeaaakeaacqGHris5daqhaaWcbaGaeGymaedabaGaeG % ymaedaaaaa!3320! \sum _1^1 $$ equivalence relations.- 33 Louveau's Theorem.- 34 Proof of Louveau's Theorem.- References.- Elephant Sandwiches....

Sommaire:
1 What are the reals, anyway?.- I On the length of Borel hierarchies.- 2 Borel Hierarchy.- 3 Abstract Borel hierarchies.- 4 Characteristic function of a sequence.- 5 Martin's Axiom.- 6 Generic G?.- 7 ?-forcing.- 8 Boolean algebras.- 9 Borel order of a field of sets.- 10 CH and orders of separable metric spaces.- 11 Martin-Solovay Theorem.- 12 Boolean algebra of order ?1.- 13 Luzin sets.- 14 Cohen real model.- 15 The random real model.- 16 Covering number of an ideal.- II Analytic sets.- 17 Analytic sets.- 18 Constructible well-orderings.- 19 Hereditarily countable sets.- 20 Shoenfield Absoluteness.- 21 Mansfield-Solovay Theorem.- 22 Uniformity and Scales.- 23 Martin's axiom and Constructibility.- 24 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfKttLearuavP1wzZbItLDhis9wBH5garm % Wu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9 % q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir % -Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGa % aeqabaWaaeaaeaaakeaacqGHris5daqhaaWcbaGaeGOmaidabaGaeG % ymaedaaaaa!3322! dollars dollars \sum _2^1 dollars dollars well-orderings.- 25 Large % aeqabaWaaeaaeaaakeaacqGHpis1daqhaaWcbaGaeGOmaidabaGaeG % ymaedaaaaa!3310! \prod _2^1 dollars dollars sets.- III Classical Separation Theorems.- 26 Souslin-Luzin Separation Theorem.- 27 Kleene Separation Theorem.- 28 % aeqabaWaaeaaeaaakeaacqGHpis1daqhaaWcbaGaeGymaedabaGaeG % ymaedaaaaa!330E! \prod _1^1 dollars dollars-Reduction.- 29 % aeqabaWaaeaaeaaakeaacqGHuoardaqhaaWcbaGaeGymaedabaGaeG % ymaedaaaaa!32E3! \Delta _1^1 dollars dollars-codes.- IV Gandy Forcing.- 30 dollars dollars equivalence relations.- 31 Borel metric spaces and lines in the plane.- 32 % aeqabaWaaeaaeaaakeaacqGHris5daqhaaWcbaGaeGymaedabaGaeG % ymaedaaaaa!3320! \sum _1^1 dollars dollars equivalence relations.- 33 Louveau's Theorem.- 34 Proof of Louveau's Theorem.- References.- Elephant Sandwiches.