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Proofs that Modify Proofs (2403.17922v1)
Published 26 Mar 2024 in math.LO
Abstract: In this paper we give an ordinal analysis of the theory of second order arithmetic. We do this by working with proof trees -- that is, "deductions" which may not be well-founded. Working in a suitable theory, we are able to represent functions on proof trees as yet further proof trees satisfying a suitable analog of well-foundedness. Iterating this process allows us to represent higher order functions as well: since functions on proof trees are just proof trees themselves, these functions can easily be extended to act on proof trees which are themselves understood as functions. The corresponding system of ordinals parallels this, using higher order collapsing function.
- “An extension of the omega-rule” In Arch. Math. Logic 55.3-4, 2016, pp. 593–603 DOI: 10.1007/s00153-016-0482-y
- Toshiyasu Arai “An ordinal analysis of ΠNsubscriptΠ𝑁\Pi_{N}roman_Π start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT-Collection”, 2024 arXiv:2311.12459 [math.LO]
- Jeremy Avigad “Ordinal analysis without proofs” In Reflections on the foundations of mathematics (Stanford, CA, 1998) 15, Lect. Notes Log. Assoc. Symbol. Logic, Urbana, IL, 2002, pp. 1–36
- Wilfried Buchholz “Relating ordinals to proofs in a perspicuous way” Essays in honor of Solomon Feferman, Papers from the symposium held at Stanford University, Stanford, CA, December 11–13, 1998 In Reflections on the foundations of mathematics (Stanford, CA, 1998) 15, Lect. Notes Log. Urbana, IL: Assoc. Symbol. Logic, 2002, pp. 37–59
- “Proof theory of impredicative subsystems of analysis” 2, Studies in Proof Theory. Monographs Bibliopolis, Naples, 1988, pp. 125
- Jean-Yves Girard “Π21subscriptsuperscriptΠ12\Pi^{1}_{2}roman_Π start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-logic. I. Dilators” In Ann. Math. Logic 21.2-3, 1981, pp. 75–219 (1982) DOI: 10.1016/0003-4843(81)90016-4
- Jean-Yves Girard “Proof theory and logical complexity” 1, Studies in Proof Theory. Monographs Bibliopolis, Naples, 1987, pp. 505
- G.E. Mints “Finite investigations of transfinite derivations” In Journal of Soviet Mathematics 10.4 Springer ScienceBusiness Media LLC, 1978, pp. 548–596 DOI: 10.1007/bf01091743
- Wolfram Pohlers “Proof theory: The First Step into Impredicativity”, Universitext Berlin, Germany: Springer, 2008
- Michael Rathjen “The art of ordinal analysis” In International Congress of Mathematicians. Vol. II Eur. Math. Soc., Zürich, 2006, pp. 45–69
- Michael Rathjen “The realm of ordinal analysis” In Sets and proofs (Leeds, 1997) 258, London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge, 1999, pp. 219–279
- Kurt Schütte “Beweistheoretische Erfassung der unerdlichen Induktion in der Zahlentheorie” In Math. Ann. 122, 1951, pp. 369–389 DOI: 10.1007/BF01342849
- Henry Towsner “Ordinal analysis by transformations” In Ann. Pure Appl. Logic 157.2-3, 2009, pp. 269–280 DOI: 10.1016/j.apal.2008.09.011
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