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ArankisanotionindescriptivesettheorythatdescribesrankssuchastheCantorBendixsonrankonthesetofclosedsubsetsofaPolishspace,differentiabilityranksontheset ofdifferentiablefunctions inC[01] suchastheKechris-Woodinrankandmanyotherranks in descriptiveset theoryandreal analysis. Thecomplexityofmanynatural ranks is 1 1 or 1 2. Weproposetounderstandthe least lengthof ranksonasetasameasureof itscomplexity. Therefore, the aimis tounderstandwhich lengths such ranksmayhave. Themain result determinesthesupremaof lengthsofcountableranksatthefirstandsecondprojectivelevels. Furthermore,wecharacterisetheexistenceofcountableranksonspecificclassesof 1 2sets.The connectionsarisingbetween 1 2 setswithcountableranksontheonehandand 1 2Borelsetson theother leadtoaconjecturethatunifiesseveral results indescriptivesettheorysuchasthe Mansfield-SolovaytheoremandarecentresultofKanoveiandLyubetsky.

Schlicht, C.P., Merlin, C., Welch, P. (2025). Countable ranks at the first and second projective levels. ISRAEL JOURNAL OF MATHEMATICS.

Countable ranks at the first and second projective levels

Schlicht, Caspar Philipp
;
2025-01-01

Abstract

ArankisanotionindescriptivesettheorythatdescribesrankssuchastheCantorBendixsonrankonthesetofclosedsubsetsofaPolishspace,differentiabilityranksontheset ofdifferentiablefunctions inC[01] suchastheKechris-Woodinrankandmanyotherranks in descriptiveset theoryandreal analysis. Thecomplexityofmanynatural ranks is 1 1 or 1 2. Weproposetounderstandthe least lengthof ranksonasetasameasureof itscomplexity. Therefore, the aimis tounderstandwhich lengths such ranksmayhave. Themain result determinesthesupremaof lengthsofcountableranksatthefirstandsecondprojectivelevels. Furthermore,wecharacterisetheexistenceofcountableranksonspecificclassesof 1 2sets.The connectionsarisingbetween 1 2 setswithcountableranksontheonehandand 1 2Borelsetson theother leadtoaconjecturethatunifiesseveral results indescriptivesettheorysuchasthe Mansfield-SolovaytheoremandarecentresultofKanoveiandLyubetsky.
2025
ISRAEL JOURNAL OF MATHEMATICS
Schlicht, C.P., Merlin, C., Welch, P. (2025). Countable ranks at the first and second projective levels. ISRAEL JOURNAL OF MATHEMATICS.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11365/1304174