Reference
J. Barwise (1975) Admissible Sets and Structures Springer-Verlag Berlin
M. Beeson (1985) Foundations of Constructive Mathematics Springer-Verlag Berlin
P. Bernays (1967) Hilbert, David, Encyclopedia of Philosophy NumberInSeries3 Macmillan and Free Press New York 496–504
E. Bishop (1967) Foundations of Constructive Analysis McGraw-Hill New York
L.E.J. Brouwer (1933) ArticleTitle‘Weten, Willen, Spreken’ (Dutch) Euclides 9 177–193
D.K. Brown (1987) ‘Functional Analysis in Weak Subsystems of Second Order Arithmetic, Ph.D. thesis Pennsylvania State University University Park, PA
D.K. Brown S. Simpson (1986) ArticleTitle‘Which Set Existence Axioms are Needed to Prove the Separable Hahn-Banach Theorem?’ Annals of Pure and Applied Logic 31 123–144 Occurrence Handle10.1016/0168-0072(86)90066-7
Buchholz W., Feferman S., Pohlers W., and Sieg W., (1981), Iterated Inductive Definitions and Subsystems of Analysis. Springer-Verlag, Berlin
H.B. Curry R. Feys (1958) Combinatory Logic NumberInSeriesI North-Holland Amsterdam
R. Diestel (1997) Graph Theory Springer-Verlag New York-Berlin-Heidelberg
M. Dummett (1973) ‘The Philosophical Basis of Intuitionistic Logic’ H.E. Rose (Eds) et al. Logic Colloquium ’73 North-Holland Amsterdam 5–40
M. Dummett (1977) Elements of Intuitionism Clarendon Press Oxford
S. Feferman (1975) ‘A Language and Axioms for Explicit Mathematics’ lecture notes in Math. 450 Springer-Verlag Berlin 87–139
S. Feferman (1979) ‘Constructive Theories of Functions and Classes’ M. Boffa D. Dalen Particlevan K. McAloon (Eds) Logic Colloquium ’78 North-Holland Amsterdam 159–224
S. Feferman (1987) ‘Proof Theory: A Personal Report’ G. Takeuti (Eds) Proof Theory EditionNumber2 North-Holland Amsterdam 445–485
S. Feferman (1988) ArticleTitle‘Hilbert’s Program Relativized: Proof-Theoretical and Foundational Reductions’ Journal of Symbolic Logic 53 364–384
S. Feferman (1998a) ‘Why a Little Bit Goes a Long Way’ S. Feferman (Eds) In the Light of Logic Oxford University Press Oxford
S. Feferman (1998) ‘Weyl Vindicated: “Das Kontinuum” 70 Years Later’ S. Feferman (Eds) In the Light of Logic Oxford University Press Oxford
Feferman S., (1996), ‘Gödel’s Program for New Axioms: Why, Where, How and What?’. Gödel ’96 Conference, Lecture Notes in Logic 6, Springer Verlag, pp. 3–22.
S. Feferman H. Friedman P. Maddy J. Steel (2000) ArticleTitle‘Does Mathematics Need New Axioms?’ Bulletin of Symbolic Logic 6 401–446 Occurrence Handle10.2307/420965
H. Friedman (1976) ArticleTitle‘Systems of Second Order Arithmetic with Restricted Induction’. I, II (abstracts) Journal of Symbolic Logic 41 557–559
H. Friedman (1977) ArticleTitle‘The Metamathematics of the Graph Minor Theorem’ Contemporary Mathematics 65 229–261
G. Gentzen (1935) ArticleTitle‘Untersuchungen über das logische Schliessen’ Mathematische Zeitschrift 39 176–210
G. Gentzen (1936) ArticleTitle‘Die Widerspruchsfreiheit der reinen Zahlentheorie’ Mathematische Annalen 112 493–565 Occurrence Handle10.1007/BF01565428
K. Gödel (1995) ‘The Present Situation in the Foundations of Mathematics’ in Collected Works NumberInSeriesIII Oxford University Press New York
Harrington L., (1977), personal communication to H. Friedman. Hilbert, D.: 1920, ‘Probleme der mathematischen Logik’. Vorlesung, Sommer-Semester 1920, Lecture Notes by Paul Bernays and Moses Schönfinkel, unpublished typescript (Bibliothek Mathematisches Institut, Universität Göttingen).
Hilbert D., (1926), ‘Über das Unendliche’. Mathematische Annalen 95, 161–190. English translation in J. van Heijenoort (eds), From Frege to Gödel. A Source Book in Mathematical Logic, (1897)–1931, Harvard University Press, Cambridge, MA, (1967).
Hilbert D., (1931), ‘Die Grundlegung der elementaren Zahlentheorie’. Mathematische Annalen 104.
D. Hilbert P. Bernays (1938) Grundlagen der Mathematik II Springer-Verlag Berlin
P. Hinman (1978) Recursion-Theoretic Hierarchies Springer-Verlag Berlin
Howard W.A., (1969), ‘The Formulae-as-Types Notion of Construction’. privately circulated notes.
Jäger G., Pohlers W., (1982), Eine beweistheoretische Untersuchung von δ1/2 –CA+ BI und verwandter Systeme, Sitzungsberichte der Bayerischen Akademie der Wissenschaften, Mathematisch–Naturwissenschaftliche Klasse.
Kreisel G., (1960), ‘Ordinal Logics and the Characterization of Informal Concepts of Proof’. in Proceedings of the 1958 International Congress of Mathematicians, Edinburgh, pp. 289–299.
Kreisel G., (1963), ‘Generalized Inductive Definitions’. in Stanford Report on the Foundations of Analysis, Section III, Mimeographed, Stanford
P. Maddy (1990) Realism in Mathematics Clarendon Press Oxford
P. Maddy (1997) Naturalism in Mathematics Clarendon Press Oxford
P. Martin-Löf (1975) ‘An Intuitionistic Theory of Types: Predicative Part’ H.E. Rose J. Sheperdson (Eds) Logic Colloquium ’73 North-Holland Amsterdam 73–118
P. Martin-Löf (1982) ‘Constructive Mathematics and Computer Programming’ L.J. Cohen J. Los H. Pfeiffer K.-P. Podewski (Eds) LMPS IV North-Holland Amsterdam
P. Martin-Löf (1984) Intuitionistic Type Theory Bibliopolis Naples
P. Martin-Löf (1987) ArticleTitle‘Truth of a Proposition, Evidence of a Judgement, Validity of a Proof’ Synthese 73 87–99
P. Martin-Löf (1995) ‘Verificationism Then and Now’ W. Pauli-Schimanovich ParticleDe E. Köhler F. Stadler (Eds) The Foundational Debate: Complexity and Constructivity in Mathematics and Physics Kluwer Academic Publishers Dordrecht
P. Martin-Löf (1996) ArticleTitle‘On the Meanings of the Logical Constants and the Justifications of the Logical Laws’ Nordic Journal of Philosophical Logic 1 11–60
J. Myhill (1975) ArticleTitle‘Constructive Set Theory’ Journal of Symbolic Logic 40 347–382
E. Nelson (1987) ‘Radically Elementary Probability Theory’ Annals of Mathematics Studies 117 Princeton University Press Princeton, NJ
E. Nelson (1993) ArticleTitle‘Taking Formalism Seriously’ The Mathematical Intelligencer 15 8–11 Occurrence Handle10.1007/BF03024251
E. Palmgren (1993) ArticleTitle‘An Information System Interpretation of Martin-Löf’s Partial Type Theory with Universes’ Information and Computation 1106 26–60
E. Palmgren (1998) ‘On Universes in Type Theory’ G. Sambin J. Smith (Eds) Twenty-five Years of Type Theory Oxford University Press Oxford 191–204
J. Paris L. Harrington (1977) ‘A Mathematical Incompleteness in Peano Arithmetic’ J. Barwise (Eds) Handbook of Mathematical Logic North Holland Amsterdam 1133–1142
C. Parsons (1970) ‘On a Number-Theoretic Choice Schema and its Relation to Induction’ A. Kino J. Myhill R.E. Vesley (Eds) Intuitionism and Proof Theory North-Holland Amsterdam 459–473
W. Pohlers (1981) ‘Proof-Theoretical Analysis of I D? by the Method of Local Predicativity’ in Iterated Inductive Definitions and Subsystems of Analysis Springer-Verlag Berlin 261–357
D. Prawitz (1977) ArticleTitle‘Meaning and Proofs: On the Conflict Between Classical and Intuitionistic Logic’ Theoria 43 11–40
M. Rathjen (1991) ArticleTitle‘Proof-Theoretic Analysis of KPM’ Archive for Mathematical Logic 30 377–403 Occurrence Handle10.1007/BF01621475
M. Rathjen (1994) ArticleTitle‘Proof Theory of Reflection’ Annals of Pure and Applied Logic 68 181–224 Occurrence Handle10.1016/0168-0072(94)90074-4
M. Rathjen (1996) ArticleTitle‘The Recursively Mahlo Property in Second Order Arithmetic’ Mathematical Logic Quarterly 42 59–66
M. Rathjen E. Palmgren (1998) ArticleTitle‘Inaccessibility in Constructive Set Theory and Type Theory’ Annals of Pure and Applied Logic 94 181–200 Occurrence Handle10.1016/S0168-0072(97)00072-9
M. Rathjen (1999) ArticleTitle‘Explicit Mathematics with the Monotone Fixed Point Principle, II: Models’ Journal of Symbolic Logic 64 517–550
Rathjen M., (2000), ‘The Superjump in Martin-Löf Type Theory’. In: Buss S., Hajek P., Pudlak P. (eds), Logic Colloquium ’98, Lecture Notes in Logic 13 (Association for Symbolic Logic) pp. 363–386.
M. Rathjen (2003) ‘The Anti-Foundation Axiom in Constructive Set Theories’ I. Loon Particlevan G. Mints R. Muskens (Eds) Games, Logic, and Constructive Sets CSLI Publications Stanford
M. Rathjen (2003) ArticleTitle‘Realizing Mahlo Set Theory in Type Theory’ Archive for Mathematical Logic 42 89–101 Occurrence Handle10.1007/s00153-002-0159-6
B. Russell (1908) ArticleTitle‘Mathematical Logic as Based on the Theory of Types’ American Journal of Mathematics 30 222–262
A. Setzer (1998) ArticleTitle‘A Well-Ordering Proof for the Proof Theoretical Strength of Martin-Löf Type Theory’ Annals of Pure and Applied Logic 92 113–159 Occurrence Handle10.1016/S0168-0072(97)00078-X
A. Setzer (2000) ArticleTitle‘Extending Martin-Löf Type Theory by One Mahlo-Universe’ Archive for Mathematical Logic 39 155–181 Occurrence Handle10.1007/s001530050140
W. Sieg (1988) ArticleTitle‘Hilbert’s Program Sixty Years Later’ Journal of Symbolic Logic 53 31–44
S. Simpson (1985) ArticleTitle‘Nichtbeweisbarkeit von gewissen kombinatorischen Eigenschaften endlicher Bäume’ Archiv f. Math. Logik 25 45–65
S. Simpson (1988) ArticleTitle‘Partial Realizations of Hilbert’s Program’ Journal of Symbolic Logic 53 349–363
S. Simpson (1999) Subsystems of Second Order Arithmetic Elsevier Science Amsterdam
W. Tait (1968) ‘Constructive Reasoning’ B. Rootselar Particlevan J.F. Staal (Eds) Logic, Methodology and Philosophy of Science III North-Holland Amsterdam 185–199
W. Tait (1981) ArticleTitle‘Finitism’ Journal of Philosophy 78 524–546
A.S. Troelstra D. Dalen Particlevan (1988) Constructivism in Mathematics: An Introduction NumberInSeriesI & II North-Holland Amsterdam
B. Waerden Particlevan der (1930) Moderne Algebra Springer Verlag Berlin
H. Weyl (1931) Die Stufen des Unendlichen Verlag von Gustav Fischer Jena
H. Weyl (1949) Philosophy of Mathematics and Natural Sciences Princeton University Press Princeton, NJ
W.H. Woodin (1994) ArticleTitle‘Large Cardinal Axioms and Independence: The Continuum Problem Revisited’ The Mathematical Intelligencer 16 IssueID3 31–35
R. Zach (1998) ArticleTitle‘Numbers and Functions in Hilbert’s Finitism’ Taiwanese Journal of Philosophy and History of Science 10 33–60
Zach R., (2001), ‘The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert’s Program’. second draft, 22 February 2001.
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Rathjen, M. The Constructive Hilbert Program and the Limits of Martin-Löf Type Theory. Synthese 147, 81–120 (2005). https://doi.org/10.1007/s11229-004-6208-4
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DOI: https://doi.org/10.1007/s11229-004-6208-4