On the Uniform Projection Problem in Descriptive Set Theory

On the Uniform Projection Problem in Descriptive Set Theory

For every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">n</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math>...

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Bibliographic Details
Main Authors: Vladimir Kanovei, Vassily Lyubetsky
Format: Article
Language:English
Published: MDPI AG 2024-12-01
Series:Axioms
Subjects:
constructibility
projective hierarchy
uniform sets
projections
Online Access:https://www.mdpi.com/2075-1680/14/1/13
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Summary:For every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">n</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>, generic models of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">ZFC</mi></semantics></math></inline-formula> will be presented for either of the following two sentences: 1. There exists a linear <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>Σ</mi><mrow><mi mathvariant="double-struck">n</mi><mo>+</mo><mn>2</mn></mrow><mn>1</mn></msubsup></semantics></math></inline-formula> set not equal to the projection of any uniform planar <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mi mathvariant="bold">Π</mi></mrow><mrow><mi mathvariant="double-struck">n</mi><mo>+</mo><mn>2</mn></mrow><mn>1</mn></msubsup></semantics></math></inline-formula> set. 2. There exists a linear <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>Δ</mi><mrow><mi mathvariant="double-struck">n</mi><mo>+</mo><mn>2</mn></mrow><mn>1</mn></msubsup></semantics></math></inline-formula> set not equal to the projection of any uniform planar <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mi mathvariant="bold">Π</mi></mrow><mrow><mi mathvariant="double-struck">n</mi><mo>+</mo><mn>1</mn></mrow><mn>1</mn></msubsup></semantics></math></inline-formula> set. Ensuing consistency and independence corollaries are discussed.
ISSN:2075-1680

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