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Gödel's program for new axioms: Why, where, how and what?

from Part I - Invited Papers

Published online by Cambridge University Press:  23 March 2017

Solomon Feferman
Stanford University
Petr Hájek
Academy of Sciences of the Czech Republic, Prague
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Summary. From 1931 until late in his life (at least 1970) Gödel called for the pursuit of new axioms for mathematics to settle both undecided number-theoretical propositions (of the form obtained in his incompleteness results) and undecided set-theoretical propositions (in particular CH). As to the nature of these, Gödel made a variety of suggestions, but most frequently he emphasized the route of introducing ever higher axioms of infinity. In particular, he speculated (in his 1946 Princeton remarks) that there might be a uniform (though non-decidable) rationale for the choice of the latter. Despite the intense exploration of the “higher infinite” in the last 30-odd years, no single rationale of that character has emerged. Moreover, CH still remains undecided by such axioms, though they have been demonstrated to have many other interesting set-theoretical consequences.

In this paper, I present a new very general notion of the “unfolding” closure of schematically axiomatized formal systems S which provides a uniform systematic means of expanding in an essential way both the language and axioms (and hence theorems) of such systems S. Reporting joint work with T. Strahm, a characterization is given in more familiar terms in the case that S is a basic system of non-finitist arithmetic. When reflective closure is applied to suitable systems of set theory, one is able to derive cardinal axioms as theorems. It is an open question how these may be characterized in terms of current notions in that subject.

Why new axioms?

Gödel's published statements over the years (from 1931 to 1972) pointing to the need for new axioms to settle both undecided number-theoretic and set-theoretic propositions are rather well known. They are most easily cited by reference to the first two volumes of the edition of his Collected Works. A number of less familiar statements of a similar character from his unpublished essays and lectures are now available in the third volume of that edition.

Given the ready accessibility of these sources, there is no need for extensive quotation, though several representative passages are singled out below for special attention.

Gödel '96
Logical Foundations of Mathematics, Computer Science and Physics - Kurt Gödel's Legacy
, pp. 3 - 22
Publisher: Cambridge University Press
Print publication year: 2017

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Aczel, P. [1980], Prege structures and the notions of proposition, truth and set, in (Barwise, J., et al, eds.) The Kleene Symposium, North-Holland, Amsterdam. 31–59.
Beeson, M. [1985], Foundations of Constructive Mathematics, Springer-Verlag, Berlin.
Bernays, P. [1961], Zur Prage der Unendlichkeitsschemata in der axiomatischen Mengenlehre, in (Bar-Hillel, Y., et al, eds.) Essays on the Foundations of Mathematics, Magnes Press, Jerusalem, 3–49. (See also Bernays [1976].)
Bernays, P. [1976], On the problem of schema of infinity in axiomatic set theory, in (Müller, G., ed.) Sets and Classes, North-Holland, Amsterdam, 121–172. (English translation of Bernays [1961].)
Feferman, S. [1962], Transfinite recursive progressions of axiomatic theories, J. Symbolic Logic 27, 259–316.Google Scholar
Feferman, S. [1964], Systems of predicative analysis, J. Symbolic Logic 29, 1–30.Google Scholar
Feferman, S. [1968], Systems of predicative analysis, II: Representations of ordinals, J. Symbolic Logic 33, 193–220.Google Scholar
Feferman, S. [1979], A more perspicuous formal system for predicativity, in (Lorenz, K., ed.) Konstruktionen vs. Positionen. Vol. I, Walter de Gruyter, Berlin, 68–93.
Feferman, S. [1988], Turing in the land of O(z), in (Herken, R., ed.) The Universal Turing Machine. A Half-century Survey, Oxford Univ. Press, Oxford, 113–147.
Feferman, S. [1991], Reflecting on incompleteness, J. Symbolic Logic 56, 1–49.Google Scholar
Feferman, S. [1991a], A new approach to abstract data types, II. Computation on ADTs as ordinary computation, in (Börger, E., et al, eds.) Computer Science Logic, Lecture Notes in Computer Science 626, 79–95.
Feferman, S. [1996], Computation on abstract data types. The extensional approach, with an application to streams, to appear in Annals of Pure and Applied Logic.
Gödel, K. [1986], Collected Works, Vol. I. Publications 1929–1936, Oxford Univ. Press, New York.
Gödel, K. [1990], Collected Works, Vol. II. Publications 1938–1974, Oxford Univ. Press, New York.
Gödel, K. [1995], Collected Works, Vol. III. Unpublished Essays and Lectures, Oxford Univ. Press, New York.
Hanf, W. P. [1964], Incompactness in languages with infinitely long expressions, Fundamenta Mathematicae 53, 309–324.Google Scholar
Hanf, W. P. and Scott, D. [1961], Classifying inaccessible cardinals (abstract), Notices A.M.S. 8, 445.Google Scholar
Jensen, R. [1995], Inner models and large cardinals, Bull. Symbolic Logic 1, 393–407.Google Scholar
Kanamori, A. [1994], The Higher Infinite, Springer-Verlag, Berlin.
Keisler, H. J. and Tarski, A. [1964], Prom accessible to inaccessible cardinals, Fundamenta Mathematicae 53, 225–308. Corrections ibid. 57 (1965) 119.Google Scholar
Kreisel, G. [1958], Ordinal logics and the characterization of informal concepts of proof, Proc. International Congress of Mathematicians (Edinburgh 1958), Cambridge Univ. Press, New York, 289–299.
Kreisel, G. [1970], Principles of proof and ordinals implicit in given concepts, in (Myhill, J., et al, eds.) Intuitionism and Proof Theory, North-Holland, Amsterdam, 489–516.
Levy, A. [1960], Axiom schemata of strong infinity in axiomatic set theory, Pacific Journal of Mathematics 10, 223–238.Google Scholar
Maddy, P. [1988], Believing the axioms, I. J. Symbolic Logic 53, 481–511.Google Scholar
Maddy, P. [1988a], Believing the axioms, II. J. Symbolic Logic 53, 736–764.Google Scholar
Martin, D. A. [1976], Hilbert's first problem: The continuum hypothesis, in (Browder, F., ed.) Mathematical Developments Arising from Hilbert's Problems, Proc. Symposia in Pure Math. 28, A.M.S., Providence, 81–92.
Moschovakis, Y. [1989], The formal language of recursion, J. Symbolic Logic 54, 1216–1252.Google Scholar
Rathjen, M. [1995], Recent advances in ordinal analysis: Π1 1 -CA and beyond, Bull. Symbolic Logic 1, 468–485.Google Scholar
Schütte, K. [1965], Eine Grenze für die Beweisbarkeit der transfiniten Induktion in der verzweigten Typenlogik, Archiv für Math. Logik und Grandlagenforschung 7, 45–60.Google Scholar
Scott, D. [1961], Measurable cardinals and constructible sets, Bull, de l'Acad. Polonaise des Sciences 9, 521–524.Google Scholar
Tait, W. [1990], The iterative hierarchy of sets, Iyyun, A Jerusalem Philosophical Quarterly 39, 65–79.Google Scholar
Tarski, A. [1962], Some problems and results relevant to the foundations of set theory, in (Nagel, E., et al, eds.) Logic, Methodology and the Philosophy of Science (Proc. of the 1960 International Congress, Stanford), Stanford Univ. Press, Stanford, 125–135.
Turing, A. [1939], Systems of logic based on ordinals, Proc. London Math. Soc., ser. 2, 45, 161–228.Google Scholar
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