On the ‘Definability of Definable’ Problem of Alfred Tarski
In this paper we prove that for any <inline-formula><math display="inline"><semantics><mrow><mi>m</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> there exists a generic extension of <...
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| Format: | Article |
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MDPI AG
2020-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/12/2214 |
| _version_ | 1797544744513961984 |
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| author | Vladimir Kanovei Vassily Lyubetsky |
| author_facet | Vladimir Kanovei Vassily Lyubetsky |
| author_sort | Vladimir Kanovei |
| collection | DOAJ |
| description | In this paper we prove that for any <inline-formula><math display="inline"><semantics><mrow><mi>m</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> there exists a generic extension of <inline-formula><math display="inline"><semantics><mi mathvariant="bold">L</mi></semantics></math></inline-formula>, the constructible universe, in which it is true that the set of all constructible reals (here subsets of <inline-formula><math display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>) is equal to the set <inline-formula><math display="inline"><semantics><msub><mi mathvariant="bold">D</mi><mrow><mn>1</mn><mi>m</mi></mrow></msub></semantics></math></inline-formula> of all reals definable by a parameter free type-theoretic formula with types bounded by <i>m</i>, and hence the Tarski ‘definability of definable’ sentence <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="bold">D</mi><mrow><mn>1</mn><mi>m</mi></mrow></msub><mo>∈</mo><msub><mi mathvariant="bold">D</mi><mrow><mn>2</mn><mi>m</mi></mrow></msub></mrow></semantics></math></inline-formula> (even in the form <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="bold">D</mi><mrow><mn>1</mn><mi>m</mi></mrow></msub><mo>∈</mo><msub><mi mathvariant="bold">D</mi><mn>21</mn></msub></mrow></semantics></math></inline-formula>) holds for this particular <i>m</i>. This solves an old problem of Alfred Tarski (1948). Our methods, based on the almost-disjoint forcing of Jensen and Solovay, are significant modifications and further development of the methods presented in our two previous papers in this Journal. |
| first_indexed | 2024-03-10T14:05:45Z |
| format | Article |
| id | doaj.art-48985eabad924552a9a8fbbae7bb9aef |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T14:05:45Z |
| publishDate | 2020-12-01 |
| publisher | MDPI AG |
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| spelling | doaj.art-48985eabad924552a9a8fbbae7bb9aef2023-11-21T00:40:22ZengMDPI AGMathematics2227-73902020-12-01812221410.3390/math8122214On the ‘Definability of Definable’ Problem of Alfred TarskiVladimir Kanovei0Vassily Lyubetsky1Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, RussiaInstitute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, RussiaIn this paper we prove that for any <inline-formula><math display="inline"><semantics><mrow><mi>m</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> there exists a generic extension of <inline-formula><math display="inline"><semantics><mi mathvariant="bold">L</mi></semantics></math></inline-formula>, the constructible universe, in which it is true that the set of all constructible reals (here subsets of <inline-formula><math display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>) is equal to the set <inline-formula><math display="inline"><semantics><msub><mi mathvariant="bold">D</mi><mrow><mn>1</mn><mi>m</mi></mrow></msub></semantics></math></inline-formula> of all reals definable by a parameter free type-theoretic formula with types bounded by <i>m</i>, and hence the Tarski ‘definability of definable’ sentence <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="bold">D</mi><mrow><mn>1</mn><mi>m</mi></mrow></msub><mo>∈</mo><msub><mi mathvariant="bold">D</mi><mrow><mn>2</mn><mi>m</mi></mrow></msub></mrow></semantics></math></inline-formula> (even in the form <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="bold">D</mi><mrow><mn>1</mn><mi>m</mi></mrow></msub><mo>∈</mo><msub><mi mathvariant="bold">D</mi><mn>21</mn></msub></mrow></semantics></math></inline-formula>) holds for this particular <i>m</i>. This solves an old problem of Alfred Tarski (1948). Our methods, based on the almost-disjoint forcing of Jensen and Solovay, are significant modifications and further development of the methods presented in our two previous papers in this Journal.https://www.mdpi.com/2227-7390/8/12/2214definability of definabletarski problemtype theoretic hierarchygeneric modelsalmost disjoint forcing |
| spellingShingle | Vladimir Kanovei Vassily Lyubetsky On the ‘Definability of Definable’ Problem of Alfred Tarski Mathematics definability of definable tarski problem type theoretic hierarchy generic models almost disjoint forcing |
| title | On the ‘Definability of Definable’ Problem of Alfred Tarski |
| title_full | On the ‘Definability of Definable’ Problem of Alfred Tarski |
| title_fullStr | On the ‘Definability of Definable’ Problem of Alfred Tarski |
| title_full_unstemmed | On the ‘Definability of Definable’ Problem of Alfred Tarski |
| title_short | On the ‘Definability of Definable’ Problem of Alfred Tarski |
| title_sort | on the definability of definable problem of alfred tarski |
| topic | definability of definable tarski problem type theoretic hierarchy generic models almost disjoint forcing |
| url | https://www.mdpi.com/2227-7390/8/12/2214 |
| work_keys_str_mv | AT vladimirkanovei onthedefinabilityofdefinableproblemofalfredtarski AT vassilylyubetsky onthedefinabilityofdefinableproblemofalfredtarski |