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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2024-12-01
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| author | Vladimir Kanovei Vassily Lyubetsky |
| author_facet | Vladimir Kanovei Vassily Lyubetsky |
| author_sort | Vladimir Kanovei |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-a31993e98b4c4eeba5621b9f0998442a |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-a31993e98b4c4eeba5621b9f0998442a2025-01-24T13:22:08ZengMDPI AGAxioms2075-16802024-12-011411310.3390/axioms14010013On the Uniform Projection Problem in Descriptive Set TheoryVladimir Kanovei0Vassily Lyubetsky1Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow 127051, RussiaInstitute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow 127051, RussiaFor 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.https://www.mdpi.com/2075-1680/14/1/13constructibilityprojective hierarchyuniform setsprojections |
| spellingShingle | Vladimir Kanovei Vassily Lyubetsky On the Uniform Projection Problem in Descriptive Set Theory Axioms constructibility projective hierarchy uniform sets projections |
| title | On the Uniform Projection Problem in Descriptive Set Theory |
| title_full | On the Uniform Projection Problem in Descriptive Set Theory |
| title_fullStr | On the Uniform Projection Problem in Descriptive Set Theory |
| title_full_unstemmed | On the Uniform Projection Problem in Descriptive Set Theory |
| title_short | On the Uniform Projection Problem in Descriptive Set Theory |
| title_sort | on the uniform projection problem in descriptive set theory |
| topic | constructibility projective hierarchy uniform sets projections |
| url | https://www.mdpi.com/2075-1680/14/1/13 |
| work_keys_str_mv | AT vladimirkanovei ontheuniformprojectionproblemindescriptivesettheory AT vassilylyubetsky ontheuniformprojectionproblemindescriptivesettheory |