Arnold Miller
Springer Berlin Heidelberg
1995e druk, 1995
9783540600596
Descriptive Set Theory and Forcing
How to prove theorems about Borel sets the hard way
Specificaties
Paperback, 133 blz.
|
Engels
Springer Berlin Heidelberg |
1995e druk, 1995
ISBN13: 9783540600596
Rubricering
Onderdeel van serie
Lecture Notes in Logic
Levertijd ongeveer 8 werkdagen
Samenvatting
An advanced graduate course. Some knowledge of forcing is assumed, and some elementary Mathematical Logic, e.g. the Lowenheim-Skolem Theorem. A student with one semester of mathematical logic and 1 of set theory should be prepared to read these notes. The first half deals with the general area of Borel hierarchies. What are the possible lengths of a Borel hierarchy in a separable metric space? Lebesgue showed that in an uncountable complete separable metric space the Borel hierarchy has uncountably many distinct levels, but for incomplete spaces the answer is independent. The second half includes Harrington's Theorem - it is consistent to have sets on the second level of the projective hierarchy of arbitrary size less than the continuum and a proof and appl- ications of Louveau's Theorem on hyperprojective parameters.
Specificaties
ISBN13:9783540600596
Taal:Engels
Bindwijze:paperback
Aantal pagina's:133
Uitgever:Springer Berlin Heidelberg
Druk:1995
Serie:Lecture Notes in Logic
Inhoudsopgave
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
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$$
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$$ well-orderings.- 25 Large
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$$ sets.- III Classical Separation Theorems.- 26 Souslin-Luzin Separation Theorem.- 27 Kleene Separation Theorem.- 28
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$$-Reduction.- 29
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$$-codes.- IV Gandy Forcing.- 30
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$$ equivalence relations.- 31 Borel metric spaces and lines in the plane.- 32
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$$
\sum _1^1
$$ equivalence relations.- 33 Louveau’s Theorem.- 34 Proof of Louveau’s Theorem.- References.- Elephant Sandwiches.
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$$
\sum _2^1
$$ well-orderings.- 25 Large
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$$
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$$ sets.- III Classical Separation Theorems.- 26 Souslin-Luzin Separation Theorem.- 27 Kleene Separation Theorem.- 28
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$$-Reduction.- 29
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$$
\Delta _1^1
$$-codes.- IV Gandy Forcing.- 30
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$$ equivalence relations.- 31 Borel metric spaces and lines in the plane.- 32
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$$
\sum _1^1
$$ equivalence relations.- 33 Louveau’s Theorem.- 34 Proof of Louveau’s Theorem.- References.- Elephant Sandwiches.