MODERN LOGIC: FROM FREGE TO GÖDEL

From Frege to GÖdel

Twentieth-century logic, and even late nineteenth-century logic, cannot be properly understood without some acquaintance not only with earlier nineteenth-century logic but also with nineteenth-century mathematics. The final section of our survey therefore begins with a sketch of the influence of nineteenth-century mathematics on the major logical developments of both the Boolean and the more recent periods. This will be followed by discussions of particular logicians.

Bibliography

(Secondary material on the logicians discussed in the final section of the article is to be found mainly in the first two parts of the bibliography.)

Carnap, Rudolf. Foundations of Logic and Mathematics. Chicago: University of Chicago Press, 1939. This is Vol. I, no. 3, of International Encyclopedia of Unified Science.

Carnap, Rudolf. Introduction to Semantics. Cambridge, MA: Harvard University Press, 1942.

Carnap, Rudolf. Meaning and Necessity. 2nd, enl. ed. Chicago: University of Chicago Press, 1956.

Church, Alonzo. Introduction to Mathematical Logic. Vol. I. Princeton, NJ: Princeton University Press, 1956.

Church, Alonzo. "A Note on the Entscheidungsproblem." Journal of Symbolic Logic 1 (1936): 40–41. See also the correction, ibid., 101–102.

Church, Alonzo. "An Unsolvable Problem of Elementary Number Theory." American Journal of Mathematics 58 (1936): 345–363.

Frege, Gottlob. Translations from the Philosophical Writings of Gottlob Frege, edited by Peter Geach and Max Black. New York: Philosophical Library, 1952.

Gentzen, Gerhard. "Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlen-theorie." Mathematische Annalen 119 (1943): 140–161.

Gentzen, Gerhard. "Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie." Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, n.s., 4 (1938): 19–44.

Gentzen, Gerhard. "Untersuchungen über das logische Schliessen." Mathematische Zeitschrift 39 (1934): 176–210, 405–431. Translated into French, with commentary, by R. Feys and J. Ladrière as Recherches sur la déduction logique. Paris, 1955.

Gentzen, Gerhard. "Die Widerspruchsfreiheit der reinen Zahlentheorie." Mathematische Annalen 112 (1936): 493–565.

Gödel, Kurt. Articles on intuitionist logic and number theory. Ergebnisse eines mathematischen Kolloquiums 4 (1933): 34–38, 40.

Gödel, Kurt. The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory. Princeton, NJ: Princeton University Press, 1940; rev. ed., 1951.

Gödel, Kurt. "What Is Cantor's Continuum Problem?" American Mathematical Monthly 54 (1947): 515–525.

Herbrand, Jacques. "Sur le Problème fondamental de la logique mathématique." Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III, 24 (1931): 12–56.

Hilbert, David. Gesammelte Abhandlungen. 3 vols. Berlin: Springer, 1932–1935.

Hilbert, David, and Paul Bernays. Grundlagen der Mathematik. 2 vols. Berlin: Springer, 1934–1939.

Post, Emil. "Formal Reductions of the General Combinatorial Decision Problem." American Journal of Mathematics 65 (1943): 197–215.

Post, Emil. "Recursively Enumerable Sets of Positive Integers and Their Decision Problems." Bulletin of the American Mathematical Society 50 (1944): 284–316.

Post, Emil. "Recursive Unsolvability of a Problem of Thue." Journal of Symbolic Logic 12 (1947): 1–11.

Quine, W. V. Set Theory and Its Logic. Cambridge, MA: Harvard University Press, 1963.

Ramsey, F. P. The Foundations of Mathematics and Other Logical Essays, edited by R. B. Braithwaite. New York: Harcourt Brace, 1931.

Skolem, Thoralf. "Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre." In Wissenschaftlicher Vorträge, gehalten auf dem Fünften Kongress der Skandinavischen Mathematiker in Helsingfors 1922,217–232. Helsinki, 1923.

Skolem, Thoralf. "Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen." In Skrifter Utgit av Videnskapsselskapet i Kristiania, I. Matematisk-naturvidenskapelig Klasse 1919, no. 4, 1–36. Oslo, 1920.

Skolem, Thoralf. "Über die Nicht-Charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen." Fundamenta Mathematicae 23 (1934): 150–161.

Skolem, Thoralf. "Über die Unmöglichkeit einer vollständigen Charakterisiergung der Zahlenreihe mittels eines endlichen Axiomsystems." Norsk Matematisk Forenings Skrifter, series 2 (10) (1933): 73–82.

Skolem, Thoralf. "Über einige Grundlagenfragen der Mathematik." In Skrifter Utgit av det Norske Videnskaps-akademi i Oslo, I. Matematisk-naturvidenskapelig Klasse, no. 4, 1–49. Oslo, 1929.

Wittgenstein, Ludwig. Philosophische Bemerkungen, edited by Rush Rhees. Oxford: Blackwell, 1964.

(A. N. P.)