(*Caveat lector*: published versions of the following may
contain some changes.)

- Degrees
of unsolvability associated with classes of formalized
theories,
*J. Symbolic Logic*22 (1957) 161-175. - The
first order properties of products of algebraic systems (with
R. L. Vaught),
*Fundamenta Mathematicae*47 (1959), 57-103. - Arithmetization
of metamathematics in a general setting,
*Fundamenta Mathematicae*49 (1960),35-92. - Transfinite
recursive progressions of axiomatic theories,
*J. Symbolic Logic*27 (1962), 259-316. - Incompleteness
along paths in progressions of theories (with C. Spector),
*J. Symbolic Logic*27 (1962), 383-390. - Classifications
of recursive functions by means of hierarchies,
*Transactions Amer. Math. Soc.*104 (1962), 101-122. - Systems
of predicative analysis,
*J. Symbolic Logic*29 (1964), 1-30. - Some
applications of the notions of forcing and generic sets,
*Fundamenta Mathematicae*56 (1965), 325-345. - Systems
of predicative analysis, II. Representation of ordinals,
*J. Symbolic Logic*33 (1968), 193-220. - Persistent and invariant formulas for outer extensions,
*Compositio Mathematica*20 (1968), 29-52. - Two
notes on abstract model theory. I. Properties invariant on the
range of definable relations between structures,
*Fundamenta Mathematicae*82 (1974) 153-165. - Two
notes on abstract model theory. II. Languages for which the set of
valid sentences is semi-invariantly implicitly definable,
*Fundamenta Mathematicae*89 (1975) 111-130. - A language and axioms for explicit mathematics, in
*Algebra and Logic*(J. N. Crossley, ed.), Lecture Notes in Mathematics 450 (1975), 87-139. - Categorical foundations and foundations of category theory, in R.E. Butts and J. Hintikka, eds.,
*Logic, Foundations of Mathematics, and Computability Theory*, Reidel, Dordrecht (1977) 149-169. - Recursion in total functionals of finite type,
*Compositio Mathematica*35 (1977), 3-22. - Recursion theory and set theory: A marriage of convenience, in
*Generalized Recursion Theory II*(J. E. Fenstad, et al., eds.), North-Holland (1978), 55-98. - Constructive theories of functions and classes, in
*Logic Colloquium '78*(M. Boffa, et al., eds.), North-Holland (1979), 159-224. - Choice
principles, the bar rule, and autonomously iterated comprehension
schemes in analysis (with G. Jäger),
*J. Symbolic Logic*48 (1983), 63-70. - Toward
useful type-free theories, I,
*J. Symbolic Logic*49 (1984), 75-111. - A theory of variable types,
*Rivista Columbiana de Matemáticas*XIX (1985), 95-105. - Hilbert's
program relativized: Proof-theoretical and foundational
reductions,
*J. Symbolic Logic*53 (1988), 364-384. - Finitary inductively
presented logics, in
*Logic Colloquium '88*(R. Ferro, et al., eds.), North-Holland, Amsterdam (1989) 191-220; reprinted in*What is a Logical System?*(D. S. Gabbay, ed.), Clarendon Press, Oxford (1994), 297-328. - Reflecting
on incompleteness,
*J. Symbolic Logic*56 (1991), 1-49. - The development of programs
for the foundations of mathematics in the first third of the 20th
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*Storia della scienza*(S. Petruccioli, ed.) Istituto della Enciclopedia Italiana, 10 v., 2001-2004, v. VIII (2004) 112-121. - Systems of explicit mathematics with non-constructive mu-operator, Part I (with G. Jäger),
*Annals of Pure and Applied Logic*65 (1993), 243-263. - Systems of explicit mathematics with non-constructive mu-operator, Part II (with G. Jäger),
*Annals of Pure and Applied Logic 79*(1996), 37-52. - What rests on what? The
proof-theoretic analysis of mathematics, in
*Philosophy of Mathematics Part I*(J. Czermak, ed.) Proc. of the 15th International Wittgenstein Symposium, Verlag Hölder-Pichler-Tempsky, Vienna (1993) 141-171; reprinted as Ch. 10 in*In the Light of Logic**,*Oxford Univ. Press, New York (1998) 187-208. - Why a little bit goes a long way:
Logical foundations of scientifically applicable mathematics,
in
*PSA 1992*, Vol. II, 442-455, 1993. Reprinted as Chapter 14 in*In the Light of Logic**,*284-298. - Predicative foundations of
arithmetic (with G. Hellman),
*J. Philosophical Logic*24 (1995) 1-17. - Godel's Dialectica interpretation
and its two-way stretch, in
*Computational Logic and Proof Theory*(G. Gottlob, et al., eds.), Lecture Notes in Computer Science 713 (1993) 23-40; reprinted as Ch. 11 in*In the Light of Logic**,*209-225. - Kreisel's "unwinding" program,
in
*Kreiseliana*(P. Odifreddi, ed.), A. K. Peters Ltd., Wellesley (1996) 247-273. - Deciding the Undecidable:
Wrestling with Hilbert's Problems, Inaugural address, Stanford
Univ., May 13, 1994, published as Ch. 1 in
*In the Light of Logic**,*3-27. - Penrose's Gödelian
argument,
*PSYCHE*2 (1996) 21-32. - Definedness,
*Erkenntnis*43 (1995) 295-320. - Computation on abstract data
types. The extensional approach, with an application to
streams,
*Annals of Pure and Applied Logic*81 (1996) 75-113. - Proof Theory Since 1960,
prepared for the
*Encyclopedia of Philosophy Supplement,*Macmillan Publishing Co., New York. - Gödel's program for new
axioms: Why, where, how and what?, in
*Gödel '96 (P. Hajek, ed.),*Lecture Notes in Logic 6 (1996), 3-22. - Challenges to predicative
foundations of arithmetic (with G. Hellman), in
*Between Logic and Intuition. Essays in Honor of Charles Parsons (*G. Sher and R. Tieszen, eds.), Cambridge Univ. Press, Cambridge (2000) 317-338. - The unfolding of non-finitist
arithmetic (with T. Strahm),
*Annals of Pure and Applied Logic*104 (2000) 75-96. - Does mathematics need new
axioms?,
*American Mathematical Monthly*106 (1999) 99-111. - My route to
arithmetization,
*Theoria*63 (1997) 168-181. - Godel's Functional
("Dialectica") Interpretation (with J. Avigad), in
*The Handbook of Proof Theory (*S. Buss, ed.), North-Holland Pub. Co., Amsterdam (1998) 337-405. - Three conceptual problems that bug me, Unpublished lecture text for 7th Scandinavian Logic Symposium, Uppsala, 1996.
- Highlights in Proof
Theory, in
*Proof Theory (*V. F. Hendricks, et al., eds.) Kluwer Academic Publishers, Dordrecht (2000) 11-31. - The significance of Hermann
Weyl's Das Kontinuum,
*ibid.*, 179-194. - Relationships between
Constructive, Predicative and Classical Systems of Analysis,
*ibid.,*221-236. - Mathematical Intuition vs.
Mathematical Monsters,
*Synthese*125*(*2000) 317-332. - Ah, Chu, in
*JFAK. Essays Dedicated to Johan van Benthem on the Occasion of his Fiftieth Birthday,*Amsterdam Univ. Press, Amsterdam (1999), CD-ROM only. - Logic, Logics, and
Logicism,
*Notre Dame J. of Formal Logic*40 (1999) 31-54. - Does reductive proof theory
have a viable rationale?,
*Erkenntnis*53 (2000) 63-96. - Alfred Tarski and a watershed
meeting in logic: Cornell, 1957 , in (J. Hintikka, et al.,
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*Philosophy and Logic. In search of the Polish tradition*, Synthese Library vol. 323, Kluwer Acad. Pubs. (2003), 151-162. - Does mathematics need new
axioms?, (Proceedings of a symposium with H. M. Friedman, P.
Maddy and J. Steel,
*Bulletin of Symbolic Logic*6 (2000) 401-413. - Tarski's conception of
logic,
*Annals of Pure and Applied Logic*126 (2004) 5-13. - Tarski's conceptual
analysis of semantical notions,
*Sémantique et épistémologie (*A. Benmakhlouf, ed.) Editions Le Fennec, Casablanca (2004) [distrib. J. Vrin, Paris] 79-108. Reprinted in (D. Patterson, ed.)*New Essays on Tarski and Philosophy*, Oxford Univ. Press (2008), 72-93. - Notes on Operational Set Theory I. Generalization of "small" large cardinals in classical and admissible set theory, draft (Theorem 4(i), p. 5, needs correction).
- Predicativity. In
*The Oxford Handbook of Philosophy of Mathematics and Logic*(S. Shapiro, ed.), Oxford University Press, Oxford (2005) 590-624. - Typical ambiguity. Trying to
have your cake and eat it too.
*One Hundred Years of Russell's Paradox*(G. Link, ed.), Walter de Gruyter, Berlin (2004) 135-151. - Some formal systems for the unlimited theory of functors and categories. Unpublished MS from 1972-73 referred to in the preceding paper, sec. 8. Uneven scanning has resulted in some missing symbols that can be restored according to context, including: p. 18, l.6, S*; p.19, Theorem 3.1, S*, and p.26, l.3, a epsilon* b.
- What kind of logic is
"Independence Friendly" logic?, in
*The Philosophy of Jaakko Hintikka*(Randall E. Auxier and Lewis Edwin Hahn, eds.); Library of Living Philosophers vol. 30, Open Court (2006), 453-469. - Comments on
"Predicativity as a philosophical position" by G. Hellman,
*Review Internationale de Philosophie (*special issue, Russell en héritage. Le centenaire des*Principles,*Ph. de Rouilhan, ed.) 229 (no. 3, 2004), 313-323. - The Gödel
editorial project: a synopsis
*Bull. Symbolic Logic*11 (2005) 132-149; reprinted in*Kurt Gödel. Essays for his Centennial*(S. Feferman, C. Parsons and S. G. Simpson, eds.),*Lecture Notes in Logic*33 (2010), Assoc. for Symbolic Logic, Cambridge University Press, 2010. - Enriched stratified systems for the
foundations of category theory, in
*What is Category Theory?*(G. Sica, ed.), Polimetrica, Milano (2006), 185-203; reprinted in (G. Sommaruga, ed.),*Foundational Theories of Classical and Constructive Mathematics*, Springer, Dordrecht (2011), 127-143. - Tarski's influence on computer science, invited lecture for LICS 2005, Chicago, June 28, 2005. Has appeared in Logical Methods in Computer Science, vol. 2 issue 3 (2006).
- Review of
*Incompleteness. The proof and paradox of Kurt Gödel,*by Rebecca Goldstein,*London Review of Books,*vol. 28, no. 3 (9 February 2006). - The impact of Gödel's incompleteness
theorems on mathematics,
*Notices American**Mathematical Society*53 no. 4 (April 2006), 434-439. - Are there absolutely unsolvable
problems? Gödel's dichotomy
*, Philosophia Mathematica,*Series III vol. 14 (2006), 134-152*.* - Turing's thesis,
*Notices American Mathematical Society*53 no. 10 (Nov. 2006), reprinted in*Alan Turing's Systems of Logic. The Princeton Thesis*(A. W. Appel, ed.), Princeton Univ. Press, 13-26. - The nature and significance of Gödel's incompleteness theorems, lecture for the Princeton Institute for Advanced Study Gödel Centenary Program, Nov. 17, 2006.
- Lieber Herr Bernays! Lieber Herr
Gödel! Gödel on finitism, constructivity and Hilbert's
, in
*Kurt Gödel and the Foundations of Mathematics. Horizons of Truth*(M. Baaz, et al., eds.) Cambridge University Press (2011), 111-133. - Lieber Herr Bernays! Lieber Herr Gödel! Gödel on finitism, constructivity and Hilbert's program, preprint of the preceding in
*Dialectica*62 (2008), 179-203. (With some editorial differences.) - Harmonious logic: Craig's interpolation theorem and its descendants, transparencies for the lecture at Interpolations--A Conference in Honor of William Craig, UC Berkeley, 13 May 2007.
- Harmonious logic: Craig's interpolation theorem and its descendants,
*Synthese*164, no. 3 (2008), 341-357. - Axioms for determinateness and
truth,
*The Review of Symbolic Logic,*1 no. 2 (2008), 204-217. - Philosophy of mathematics: 5
questions, in
*Philosophy of Mathematics: 5 Questions*(V. F. Hendricks and H. Leitgeb, eds.), Automatic Press/VIP 2008, 115-135. - Gödel, Nagel, minds and
machines, Ernest Nagel Lecture, Columbia University, Sept. 27,
2007;
*J. Philosophy*CVI, nr. 4, April 2009, 201-219. - The Proof Theory of Classical and Constructive Inductive Definitions: A 40 year saga, slides for an invited talk at the Pohlersfest, Münster 18 July 2008.
- Conceptual structuralism and the continuum, slides for talk Phil Math Intersem 2010 Université Paris 7-Diderot, 6/08/10 (revision of slides for talk at VIIIth International Ontology Congress, San Sebastián, 10/01/08).
- Conceptions of the continuum, slides for talk at Barcelona Workshop on the Foundations of Mathematics, October 6, 2008
*.* - And so on ...: Reasoning with infinite diagrams, slides for talk at Workshop on Diagrams in Mathematics, Paris, October 9, 2008, and Logic Seminar, Stanford, February 24, 2009.
- What's definite? What's not? Slides for talk at Harvey Friedman 60th birthday conference, Ohio State U., May 16, 2009.
- On the strength of some semi-constructive theories, in
*Proof, Categories and Computation: Essays in honor of Grigori Mints*, (S. Feferman and W. Sieg, eds.), College Publications, London (2010), 109-129; reprinted in*Logic, Construction, Computation*(U. Berger, et al., eds.)(H. Schwichtenberg Festschrift volume), Ontos Verlag, Frankfurt (2012), 201-225. - The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968-2008, in (R. Schindler, ed.)
*Ways of Proof Theory*, Ontos Verlag, Frankfurt (2010), 7-30. - Operational set theory and small large cardinals,
*Information and Computation*207 (2009), 971-979. - Modernism in mathematics, review of
*Plato's Ghost*by Jeremy Gray (Princeton U. Press, 2008),*American Scientist*97 no. 5 (Sept-Oct 2009), 417. - Conceptions of the continuum,
*Intellectica*51 (2009/1), 169-189. - The unfolding of finitist arithmetic (with Thomas Strahm),
*The Review of Symbolic Logic*3 (2010), 665-689*.* - Set-theoretical invariance criteria for logicality,
*Notre Dame J. of Formal Logic*51 (2010), 3-20. - Gödel's incompleteness theorems, free will and mathematical thought, in
*Free Will and Modern Science*(R. Swinburne, ed.), OUP for the British Academ (2011), 102-122. - Foundations of category theory: What remains to be done, slides for contributed talk at ASL 2011 meet, UC Berkeley, March 24, 2011.
- Axiomatizing Truth: How and Why , slides for invited lecture at Pillars of Truth Conference, Princeton University, April 8-10, 2011.
- Axiomatizing truth: Why and how, in
*Logic, Construction, Computation*(U. Berger, et al., eds.)(H. Schwichtenberg Festschrift volume), Ontos Verlag, Frankfurt (2012) 185-200. - About and around computing over the reals, in
*Computability: Gödel, Church, Turing and Beyond*(J. Copeland, C. Posy and O. Shagrir, eds), MIT Press (2013), 55-76. - Which quantifiers are logical?: A combined semantical and inferential criterion, slides for talk Aug 08, 2011, ESSLLI Workshop on Logical Constants, Ljubljana.
- Which quantifiers are logical? A combined semantical and inferential criterion, text for preceding talk; to appea in
*Synthese.*. - Is the Continuum Hypothesis a definite mathematical problem?, Draft of paper for the lecture to the Philosophy Dept., Harvard University, Oct. 5, 2011 in the EFI project series.
- Is the Continuum Hypothesis a definite mathematical problem?, slides for the preceding talk.
- Turing's 'Oracle': From absolute to relative computability--and back, slides for Logic Seminar talk, Stanford, April 10, 2012.
- About and around computing over the reals, slides for Logic Seminar talk, Stanford, April 17, 2012.
- And so on... Reasoning with infinite diagrams,
*Synthese*186, no. 1 (2012), 371-386. - Review of Curtis Franks
*The Autonomy of Mathematical Knowledge. Hilbert's program revisited**,**Philosophia Mathematica. Series III*, 20 no.3 (2012), 387-400. - On rereading van Heijenoort's
*Selected Essays*,*Logica Universalis*6 no. 3-4 (2012), 535-552. - Introduction to
*Foundations of Explicit Mathematics* - Three Problems for Mathematics; Lecture 1: Bernays, Gödel, and Hilbert's consistency program, slides for inaugural Paul Bernays Lectures, ETH, Zurich, Sept. 11, 2012.
- Three Problems for Mathematics: Lecture 2: Is the Continuum Hypothesis a definite mathematical problem?, slides for inaugural Paul Bernays Lectures, ETH, Zurich, Sept. 12, 2012.
- Three Problems for Mathematics; Lecture 3: Foundations of Unlimited Category Theory, slides for inaugural Paul Bernays Lectures, ETH, Zurich, Sept. 12, 2012.
- What's special about mathematical proofs?, Remarks for the Williams Symposium on Proof, University of Pennsylvania, Nov. 9, 2012.
- Foundations of unlimited category theory: What remains to be done,
*The Review of Symbolic Logic*6 (2013), 6-15. - Why isn't the Continuum Problem on the Millennium ($1,000,000) Prize list?, slides for CSLI Workshop on Logic, Rationality and Intelligent Interaction, Stanford, June 1, 2013.
- How a little bit goes a long way: Predicative foundations of analysis, unpublished notes dating from 1977-1981, with a new introduction.
- Theses for computation and recursion on concrete and abstract structures, to appear in a Turing Centennial volume for Birkhäuser edited by G. Sommaruga and T. Strahm.
- Categoricity and open-ended axiom systems, lecture slides for the conference, "Intuition and Reason", in honor of Charles Parsons, Tel-Aviv, Dec. 2, 2013; full YouTube video.
- The operational perspective, lecture slides for the conference, "Advances in Proof Theory 2013" in honor of Gerhard Jäger, Bern, Dec. 14, 2013.
- Logic, mathematics and conceptual structuralism, in
*The Metaphysics of Logic*(P. Rush, ed.), Cambridge University Press, 2014 (to appear). - A fortuitous year with Leon Henkin, to appear in
*The Life and Work of Leon Henkin--Essays on his Contributions*(M. Manzano, I. Sain and E. Alonso, eds.), Springer International., - The Continuum Hypothesis is neither a definite mathematical problem nor a definite logical problem, revised version of 2011 Harvard Philos. Dept. EFI lecture.
- The operational perspective: Three routes, to appear in a Festschrift for the 60th birthday of Gerhard Jäger.
- In Memoriam: Grigori E. Mints, 1939-2014, to appear in
*The Bulletin of Symbolic Logic.*