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Abstract: In [15],[16] Kreisel introduced the no-counterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated "-substitution method (due to W. Ackermann), that for every theorem A (A prenex) of first-order Peano arithmetic PA one can find ordinal recursive functionals \Phi A of order type ! " 0 which realize the Herbrand normal form A of A. (Update)
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.... in GnR [B 0;1 ] instead of GnR ) and its verification can be carried out in GnA Delta (BR 0;1 ) Pi 0 1 (DC 0 ) By [16] (proposition 4.2) it follows (since deg(fl1) 2) that Phi can be written as a primitive recursive functional Phi such that PA BR...
...by Spector [118] an which goes beyond Godel s primitive recursive functionals. We will discuss this further in chapters to come (see also [82] for a thorough discussion of the modus ponens problem for the no counterexample interpretation) For the time being we confine...
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BibTeX entry: (Update)
Kohlenbach, U., On the no-counterexample interpretation. J. Symbolic Logic 64, pp. 1491-- 1511 (1999). 23 http://citeseer.ist.psu.edu/kohlenbach97nocounterexample.html More
@misc{ kohlenbach99nocounterexample,
author = "U. Kohlenbach",
title = "the no-counterexample interpretation",
text = "Kohlenbach, U., On the no-counterexample interpretation. J. Symbolic Logic
64, pp. 1491-- 1511 (1999). 23",
year = "1999",
url = "citeseer.ist.psu.edu/kohlenbach97nocounterexample.html" }
Citations (may not include all citations):104 Metamathematical investigation of intuitionistic arithmetic .. (context) - Troelstra
49 Recursive functionals and quantifiers of finite types (context) - Kleene - 1959
40 Iterated Inductive Definitions and Subsystems of Analysis: R.. (context) - Buchholz, Feferman et al. - 1981
30 Theories of finite type related to mathematical practice (context) - Feferman
30 Proof theory: some applications of cut-elimination (context) - Schwichtenberg
21 Normal derivability in classical logic (context) - Tait - 1968
18 Provably recursive functionals of analysis: a consistency pr.. (context) - Spector - 1962
14 Reflection principles and their use for establishing the com.. (context) - Kreisel, Levy - 1968
12 the interpretation of non-finitist proofs (context) - Kreisel - 1951
12 A survey of proof theory (context) - Kreisel - 1968
10 Mathematical significance of consistency proofs (context) - Kreisel - 1958
7 Functional interpretation of bar induction by bar recursion (context) - Howard - 1968
6 part II: Interpretation of number theory (context) - Kreisel - 1952
6 quantifier induction (context) - Parsons - 1972
5 Proof-theoretic analysis of restricted induction schemata (context) - Parsons - 1971
5 Transfinite induction and bar induction of types zero and on.. (context) - Howard, Kreisel - 1966
5 A model for bar recursion of higher types (context) - Scarpellini
4 Unwinding Program (context) - Feferman - 1996
4 On bar recursion of types 0 and (context) - Schwichtenberg - 1979
4 The substitution method (context) - Tait - 1965
3 Ordinal analysis of simple cases of bar recursion (context) - Howard - 1981
2 Ordinal analysis of bar recursion of type zero (context) - Howard
2 Unpublished Essays and Lectures (context) - Godel - 1995
1 With English translation in [8 (context) - Godel - 1938
1 Minimal realizability (context) - Damnjanovic - 1995
1 Vortrag bei Zilsels (context) - Sieg, Parsons
1 Einige Anwendungen von unendlichen Termen und Wertfunktional.. (context) - Schwichtenberg - 1973
1 Zur Widerspruchsfreiheit der reinen Zahlentheorie (context) - Ackermann - 1940
1 Minimal Realizability (context) - Kohlenbach, on - 1995
1 Uber eine bisher noch nicht benutze Erweiterung des finiten .. (context) - Godel - 1958
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