Reverse Mathematik jenseits der Gödelhierarchie

• Antragsteller: Dr. Sam Sanders
• Fachliche Zuordnung: Mathematik
• Förderung: Förderung von 2019 bis 2023
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 423971947

• Erstellungsjahr: 2024

Zusammenfassung der Projektergebnisse

Das Ziel meines Projekts war es, neue Ergebnisse in der Reverse Mathematics zu entwickeln, die den Stand der Technik, wie er in der Gödel-Hierarchie verkörpert ist, in Frage stellen. Zu den Hauptergebnissen meines Projekts gehören zwei wichtige Erweiterungen des bekannten Big-Five-Phänomens der Reverse Mathematics, wobei eine Erweiterung den Stand der Technik in Frage stellt. Weitere wichtige Erkenntnisse sind ein wichtiger Zusammenhang zwischen reeller und hyperarithmetischer Analyse und die Identifizierung von Grundprinzipien, wie dem Supremumprinzip für stetige Funktionen, die Arithmetik zweiter Ordnung in ihren verschiedenen Erscheinungsformen implizieren. Letzteres stellt den Stand der Technik erneut in Frage.

Projektbezogene Publikationen (Auswahl)

• Reverse formalism 16. Synthese, 197(2), 497-544.
  Sanders, Sam
  
• COMPUTABILITY THEORY, NONSTANDARD ANALYSIS, AND THEIR CONNECTIONS. The Journal of Symbolic Logic, 84(4), 1422-1465.
  Normann, Dag & Sanders, Sam
  
• Lifting Recursive Counterexamples to Higher-Order Arithmetic. Lecture Notes in Computer Science, 249-267. Springer International Publishing.
  Sanders, Sam
  
• Nets and Reverse Mathematics. Lecture Notes in Computer Science, 253-264. Springer International Publishing.
  Sanders, Sam
  
• Reverse Mathematics and Computability Theory of Domain Theory. Logic, Language, Information, and Computation, 550–568.
  Sanders, Sam
  
• The strength of compactness in Computability Theory and Nonstandard Analysis. Annals of Pure and Applied Logic, 170(11), 102710.
  Normann, Dag & Sanders, Sam
  
• Open sets in computability theory and reverse mathematics. Journal of Logic and Computation, 30(8), 1639-1679.
  Normann, Dag & Sanders, Sam
  
• Pincherle's theorem in reverse mathematics and computability theory. Annals of Pure and Applied Logic, 171(5), 102788.
  Normann, Dag & Sanders, Sam
  
• Reverse Mathematics of Topology: Dimension, Paracompactness, and Splittings. Notre Dame Journal of Formal Logic, 61(4).
  Sanders, Sam
  
• Splittings and Disjunctions in Reverse Mathematics. Notre Dame Journal of Formal Logic, 61(1).
  Sanders, Sam
  
• The unreasonable effectiveness of Nonstandard Analysis. Journal of Logic and Computation, 30(1), 459-524.
  Sanders, Sam
  
• Between Turing and Kleene. Lecture Notes in Computer Science, 281-300. Springer International Publishing.
  Sanders, Sam
  
• Betwixt Turing and Kleene. Lecture Notes in Computer Science, 236-252. Springer International Publishing.
  Normann, Dag & Sanders, Sam
  
• Countable sets versus sets that are countable in reverse mathematics. Computability, 11(1), 9-39.
  Sanders, Sam
  
• Splittings and Robustness for the Heine-Borel Theorem. Lecture Notes in Computer Science, 395-406. Springer International Publishing.
  Sanders, Sam
  
• The Axiom of Choice in computability theory and Reverse Mathematics with a cameo for the Continuum Hypothesis. Journal of Logic and Computation, 31(1), 297-325.
  Normann, Dag & Sanders, Sam
  
• Lifting proofs from countable to uncountable mathematics. Information and Computation, 287, 104762.
  Sanders, Sam
  
• ON ROBUST THEOREMS DUE TO BOLZANO, WEIERSTRASS, JORDAN, AND CANTOR. The Journal of Symbolic Logic, 89(3), 1077-1127.
  Normann, Dag & Sanders, Sam
  
• On the computational properties of basic mathematical notions. Journal of Logic and Computation, 32(8), 1747-1795.
  Normann, Dag
  
• ON THE UNCOUNTABILITY OF. The Journal of Symbolic Logic, 87(4), 1474-1521.
  Normann, Dag & Sanders, Sam
  
• On the Computational Properties of the Uncountability of the Real Numbers. Lecture Notes in Computer Science, 362-377. Springer International Publishing.
  Sanders, Sam
  
• Representations and the Foundations of Mathematics. Notre Dame Journal of Formal Logic, 63(1).
  Sanders, Sam
  
• Reverse Mathematics of the Uncountability of R. Lecture Notes in Computer Science, 272-286. Springer International Publishing.
  Sanders, Sam
  
• BIG IN REVERSE MATHEMATICS: MEASURE AND CATEGORY. The Journal of Symbolic Logic, 90(4), 1593-1638.
  Sanders, Sam
  
• BIG IN REVERSE MATHEMATICS: THE UNCOUNTABILITY OF THE REALS. The Journal of Symbolic Logic, 1-34.
  Sanders, Sam
  
• Historical Infinitesimalists and modern historiography of infinitesimals. Antiquitates Mathematicae, 16(1).
  Katz, Mikhail; Bair, Jacques; Borovik, Alexandre; Kanovei, Vladimir; Kutateladze, Semen S.; Sanders, Sam; Sherry, David & Ugaglia, Monica
  
• Is Pluralism in the History of Mathematics Possible?. The Mathematical Intelligencer, 45(1), 8-8.
  Bair, Jacques; Borovik, Alexandre; Kanovei, Vladimir; Katz, Mikhail G.; Kutateladze, Semen S.; Sanders, Sam; Sherry, David; Ugaglia, Monica & van Atten, Mark
  
• The Biggest Five of Reverse Mathematics. Journal of Mathematical Logic, 25(01).
  Normann, Dag & Sanders, Sam
  
• The Non-normal Abyss in Kleene’s Computability Theory. Lecture Notes in Computer Science, 37-49. Springer Nature Switzerland.
  Sanders, Sam
  
• The Vitali covering theorem in Reverse Mathematics and computability theory, To appear in Annals of Pure and Applied Logic.
  Normann, Dag & Sanders, Sam
  
• A NOTE ON CONTINUOUS FUNCTIONS ON METRIC SPACES. The Bulletin of Symbolic Logic, 30(3), 398-420.
  Sanders, Sam
  
• APPROXIMATION THEOREMS THROUGHOUT REVERSE MATHEMATICS. The Journal of Symbolic Logic, 1-32.
  Sanders, Sam
  
• Coding is hard.
  Sanders, Sam
  
• Connecting real and hyperarithmetical analysis. Documenta Mathematica, 29(6), 1469-1498.
  Sanders, Sam
  
• On the logical janus face of weak continuity
  Sanders, Sam