Reverse Mathematics beyond the Gödel hierarchy

• Applicant: Dr. Sam Sanders
• Subject Area: Mathematics
• Term: from 2019 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 423971947

• Final Report Year: 2024

Final Report Abstract

The aim of my project was to develop new results in Reverse Mathematics that challenge the state-of-the-art as embodied by the Godel hierarchy. The main findings of my project include two major extensions of the well-known Big Five phenomenon of Reverse Mathematics, with one of the two extensions challenging the state-of-the-art. Other important findings are a major connection between real and hyperarithmetical analysis and the identification of basic principles, like the supremum principle for continuous functions on metric spaces, that imply second-order arithmetic in its various guises; he latter again challenges the state-of-the-art.

Publications

• 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