Abstract
This paper is a summary of a more comprehensive work Infinitarna Logika Ernsta Zermela (The Infinitary Logic of Ernst Zermelo) being currently under preparation for the research grant KBN 2H01A 00725 Metody nieskończonościowe w teorii definicji (Infinitary methods in the theory of definitions) headed by Professor JANUSZ CZELAKOWSKI at the Institute of Mathematics and Information Science of the University of Opole, Poland. The presentation of Zermelo's ideas is accompanied with some remarks concerning the development of infinitary logic.References
Ajdukiewicz, K. 1928. Główne zasady metodologji nauk i logiki formalnej. (Skrypt autoryzowany, zredagowany przez Mojżesza Presburgera.) Warszawa: Wydawnictwa Koła Matematyczno-Fizycznego Słuchaczów Uniwersytetu Warszawskiego.
Barwise, J. 1974. Axioms for abstract model theory. Annąls of Mathematical Logic, vol. 7, 221-265.
Barwise, J. 1975. Admissible Sets and Structures. An Approąch to Definąbility Theory. Berlin Heidelberg New York: Springer Verlag.
Barwise, J., Feferman, S. (Eds.) 1985. Model-Theoretic Logics. New York Berlin Heidelberg Tokyo: Springer Verlag.
Batóg, T. 20002. Dwa paradygmaty matematyki. Studium z dziejów i filozofii matematyki. Poznan: Wydawnictwo Naukowe UAM.
Bell, J.L. 2004. Infinitary Logic. StanfordEncyclopedią of Philosophy.
Benacerraf, P., Wright, C. 1985. Skolem and the Skeptic. Proceedings of the Aristoteliąn Society, Supplementary Volume 59, 85-137. Reprinted in: [Shapiro 1996].
Bays, T. 2000. Reflections on Skolem 's Pąrądox. PhD Thesis.
Buldt, B. 2002. Kompaktheit und Endlichkeit in der formalen Logik. In: B. Buldt u.a. (Hrsg.) Kurt Godel. Wąhrheit und Beweisbąrkeit. Band 1: Dokumente und historische Anąlysen, Band 2: Kompendium zum Werk. Wien: obv&hpt VerlagsgmbH & Co., 31-49.
van Dalen, D., Ebbinghaus, H.D. 2000. Zermelo and the Skolem Paradox. The Bulletin of Symbolic Logic Volume 6, Number 2 145-161.
Dawson, J.W. 1985. Completing the Godel-Zermelo Correspondence. Historia Mąthemąticą 12, 66-70.
DeLong, H. 1970. A Profile of Mathematical Logic. Addison-Wesley Publishing Company.
Dickmann, M.A. 1975. Ląrge Infinitary Ląnguąges. Amsterdam: North Holland.
Ebbinghaus, H.D. 2001. Zermelo in the Mirror of the Baer Correspondence 1930-1931. Submitted for Historią Mąthemąticą. Na stronie: http://hrz.upb.de/~apeck1/zermelo/id9.htm
Ebbinghaus, H.D. 2003. Zermelo: On boundary numbers and domains of sets, Part II? Na stronie: http://hrz.upb.de/~apeck1/zermelo/id9.htm
Ebbinghaus, H.D. 2003a. Zermelo: Definiteness and the Universe ofDefinable Sets. History and Philosophy of Logic 24. 197-219.
Ferreirós, J. 2001. The road to modern logic — an interpretation. The Bulletin of Symbolic Logic Volume 7, Number 4, 441-484.
Fraenkel, A. 19232. Einleitung in die Mengenlehre. Berlin.
Fraenkel, A., Bar Hillel, Y., Levy, A. 1973. Foundątions of set theory. Amsterdam - London: North Holland Publishing Company.
George, A. 1985. Skolem and the Lowenheim-Skolem Theorem: A Case Study of the Philosophical Significance of Mathematical Results. History and Philosophy of Logic, 6, 75-89. Reprinted in: [Shapiro 1996].
Grattan-Guinness, I. 1979. In memoriam Kurt Godel: his 1931 correspondence with Zermelo on his incompletability theorem. Historią Mąthemąticą 6, 294-304.
Hallet, M. 1984. Cąntoriąn set theory and limitątion of size. Oxford: Clarendon Press.
van Heijenoort, J. (Ed.) 1967. From Frege to Godel: A source book in mathematical logic, 1879-1931. Cambridge, Mass.
Helmer, O. 1938. Languages with expressions of infinite length. Erkenntnis 8, 138-141.
Henkin, L. 1955. The Representation Theorem for Cylindrical Algebras. In: Mathematical Interpretation of Formąl Systems. North Holland, 85-97.
Henkin, L. 1959. Some remarks on infinitely long formulas. In: Infinitistic Methods. Proceedings of the Symposium on Foundątions of Mąthemątics. Wąrsąw, 2-9 September 1959. Oxford London New York Paris: Pergamon Press; Warszawa: Panstwowe Wydawnictwo Naukowe, 167-183.
Hilbert, D., Bernays, P. 1934 (I), 1939 (II). Grundlągen der Mathematik. Berlin.
Hunter, G. 1971. Metąlogic. An Introduction to the Metątheory of Standard First Order Logic. Macmilan Press.
Hunter, G. 1982. Metąlogiką. Warszawa: PWN.
Jordan, P. 1949. Zur Axiomatik der Verknupfungsbereiche. Abhąnd. Mąth. Sem. Hamburg. Univ 16, 54-70.
Kanamori, A. 1996. The mathematical development of set theory from Cantor to Cohen. The Bulletin of Symbolic Logic Volume 2, Number 1, 1-71.
Kanamori, A. 2004. Zermelo and set theory. The Bulletin of Symbolic Logic Volume 10, Number 4, 487-553.
Karp, C. 1964. Ląnguąges with Expressions of Infinite Length. Amsterdam: North Holland.
Keisler, H.J. 1971. Model Theory for Infinitary Logic. Amsterdam: North Holland.
Keisler, H.J., Knight, J.L. 2004. Barwise: infinitary logic and admissible sets. The Bulletin of Symbolic Logic Volume 10, Number 1, 4-36.
Klenk, V. 1976. Intended Models and the Lowenheim-Skolem Theorem. Journal of Philosophicąl Logic, 5, 475-489. Reprinted in: [Shapiro 1996].
Krasner, M. 1938. Une generalisation de la notion de corps. Journal de Mąthemątiques Pures et Appliąuees 9, 367-385.
Kripke, S. 1964. Transfinite recursion on admissible ordinals, I, II (abstracts). Journal of Symbolic Logic 29, 161-162.
Krynicki, M., Mostowski, M., Szczerba, L. (Eds.) 1995. Quąntifiers, logics, models and computątion. Kluwer Academic Publishers.
Kuratowski, K. 1937. Les types d’ordre definissables et les ensembles boreliens. Fundamenta Mathematicae 29, 97-100.
Lewis,C.S. 1918. A Survey of Symbolic Logic. Berkeley: University of California.
Lindenbaum, A., Tarski, A. 1936. Uber die Beschrankheit der Ausdruckmittel deduktiver Theorien. Ergebnisse eines mąthemątischen Kolloquiums 7, 1934-1935, 15-22.
Lindstrom, P. 1966. First order predicate logic with generalized quantifiers. Theorią, 32, 186-195.
Lindstrom, P. 1969. On Exteńsiońs of Elementary Logic. Theorią, 35, 1-11.
Lowenheim, L. 1915. Uber Moglichkeiten im Relativkalkul. Mąthemątische Annąlen, 68, 169-207. Translated and reprinted in: ^an He^jenoort 1967], 228-251.
McCarthy, Ch., Temant, N. 1987. Skolem’s Paradox and Constructivism. Journal of Philosophicąl Logic, 16, 165-202. Reprinted in: [Shapiro 1996].
Moore, A.W. 1985. Set Theory, Skolem’s Paradox and the Trąctątus. Anąlysis, 45, 13-20.
Moore, G.H. 1980. Beyond First-order Logic: The Historical Interplay between Mathematical Logic and Axiomatic Set Theory. History and Philosophy of Logic, 1, 95-137. Reprinted in: [Shapiro 1996].
Moore, G.H. 1982. Zermelo 's ąxiom of choice. Its origins, development and influence. New York, Heidelberg, Berlin.
Moore, G.H. 1995. The prehistory of infinitary logic: 1885-1955. In: Maria Luisa Dalla Chiara, Kees Doets, Daniele Mundici, Johan van Benthem (Eds.) Structures and norms in science. Volume two of the Tenth International Congress of
Logic, Methodology and Philosophy of Science, Florence, August 1995, Kluwer Academic Publishers, 105-123.
Moore, G.H. 2002. Die Kontroverse zwischen Godel und Zermelo. In: B. Buldt u.a. (Hrsg.) Kurt Gódel. Wąhrheit und Beweisbarkeit. Band 1: Dokumente und historische Anąlysen, Band 2: Kompendium zum Werk. Wien: obv&hpt VerlagsgmbH & Co., 55-64.
Mostowski, A. 1948. Logika matematyczna. Warszawa-Wroclaw.
Mostowski, A. 1957. On a generalization of quantifiers. Fundamenta Mąthemąticąe, 44, 12-36.
Murawski, R. 20012. Filozofia matematyki. Zarys dziejów. Warszawa: Wydawnictwo Naukowe PWN.
Murawski, R. 2002. Współczesna filozofia matematyki. Warszawa: Wydawnictwo Naukowe PWN.
Murawski, R. 20033. Filozofia matematyki. Antologia tekstów klasycznych. Poznan: Wydawnictwo Naukowe UAM.
Myhill, J. 1951. On the ontological significance of the Lowenheim-Skolem theorem. In: M. White (Ed.) Academic Freedom, Logic and Religion. The University of Pe^sy^nia Press, 57-70.
Myhill, J. 1952. The hypothesis that all classes are nameable. Proc. Nąt. Acąd. Sci. USA, 38, 979.
von Neumann, J. 1925. Eine Axiomatisierung der Mengenlehre. Journal fur die reine und ąngewąndte Mathematik, 154, 219-240. Translated and reprinted in: [van Heijenoort 1967].
Peckhaus, V. 1990. ‘Ich habe mich wohl gehutet, alle Patronen auf einmal zu verschieBeń’. Ernst Zermelo in Gottingen. History and Philosophy of Logic 11, 19-58.
Peckhaus, V. 200?. Pro and Contra Hilbert: Zermelo’s Set Theories. Na stronie: http://hrz.upb.de/~apeck1/zermelo/id9.htm
Peirce, C.S. 1885. On the algebra of logic: a contribution to the philosophy of notation. American Journal of Mathematics 7, 180-202.
Platek, R. 1966. Foundątions of recursion theory. Doctoral Dissertation and Supplement. Stanford: Stanford Umversity.
Putnam, H. 1980. Models and Reality. Journal of Symbolic Logic, 45, 464-482.
Quine, W.V.O. 1966. Ontological Reduction and the World of Numbers. In: W.V.O. Quine The Ways of Paradoxand Other Essąys. New York: Random House, 199-207.
Resnik, M. 1966. On Skolem’s Paradox. Journal of Philosophy Volume 63 Number 15, 425-438.
Resnik, M. 1969. More on Skolem’s Paradox. Nous, 3, 185-196.
Robinson, A. 1951. On the Metąmąthemątics of Algebra. Amsterdam: North Holland.
Robinson, A. 1957. Applications to Field Theory. Summaries oftalks at the Summer Institute for Symbolic Logic in 1957 at Cornell University, 326-331.
Schroder, E. 1895. Vorlesungen uber die Algebra der Logik. Vol. 3, Leipzig.
Schroder, E. 1910. Abriss der Algebra der Logik, 2, E. Muller (Ed.), Leipzig.
Scott, D., Tarski, A. 1957. The sentential calculus with mfimtely long expressions. Summąries of tąlks ąt the Summer Institute for Symbolic Logic in 1957 ąt Cornell University, 83-89.
Scott, D., Tarski, A. 1958. The sentential calculus with mfimtely long expressions. Colloquium Mąthemąticum 6,166-170.
Shapiro, S. (Ed.) 1996. Thelimits of logic: higher-order logic and the Lówenheim-Skolem theorem. Aldershot: Dartmouth Publishing Company.
Skolem, T. 1919. Uńtersuchuńgeń uber die Axiome des Klassenkalkuls und uber Produktations- und Summationsprobleme, welche gewisse Klassen von Aussagen betreffen. Videnskąpsselskąpets skrifter, I. Mątemątisk-nąturvedenskąbelig klasse, no 3.
Skolem, T. 1920. Logisch-kombinatorische Uńtersuchuńgeń uber die Erfullbarkeit oder Beweisbarkeit mathematischer Satze nebst einem Theoreme uber dichte Mengen. Videnskąppselskąpets skrifter, I. Mątemątisk-nąturvedenskąbelig klasse, no 4. Translated and reprinted in: [yań Heijenoort 1967], 252-263.
Skolem, T. 1922. Einige Bemerkungen zur axiomatischeń Begrundung der Mengenlehre. Mątemątikerkongressen i Helsingfors den 4-7 Juni 1922, Den femte skąndinąviską mątemątikerkongressen, Redogórelse, (Akademiska Bokhandeln, Helsinki, 1923).
Translated and reprinted in: [yań Heijenoort 1967], 290-301.
Skolem, T. 1930. Einige Bemerkungen zu der Abbandlung von E. Zermelo “Uber die Definitheit in der Axiomatik”. Fundamenta Mathematicae 15, 337-341.
Suszko, R. 1951. Canonic axiomatic systems. Studia Philosophicą, IV, 301-330.
Tarski, A. 1933. Pojęcie prawdy wjęzykach nauk dedukcyjnych. Towarzystwo Naukowe Warszawskie, Warszawa.
Tarski, A. 1934. Z badan metodologicznych nad definiowalnościa terminów. Przegląd Filozoficzny 37, 438-460.
Tarski, A. 1986. What are logical notions? History and Philosophy of Logic, 7, 143-154.
Taylor, R.G. 1993. Zermelo, Reductionism, and the Philosophy of Mathematics. Notre Dąme Journal of Formal Logic Volume 34, Number 4, 539-563.
Taylor, R.G. 2002. Zermelo’s Cantorian theory of systems of mfimtely long propositions. The Bulletin of Symbolic Logic Volume 8, Number 4, 478-515.
Thomas, W. 1971. On behalf of the Skolemite. Anąlysis 31, 177-186.
Thomas, W. 1968. Platonism and the Skolem Paradox. Analysis, 28, 193-196.
Wang, H. 1955. On denumerable bases offormal systems. In: Mathematicalinterpretation of formalsystems. Amsterdam: North Holland Publishing Company.
Wang, H. 1962. A survey of mathematical logic. Pekińg: Science Press, Amsterdam: North Holland Publishińg Company.
Westerstahl, D. 1989. Quańtifiers in formal and ńatural lańguages. In: D. Gabbay, F. Gueńthńer (Eds.) Handbook of Philosophical Logic, vol. IV, D. Reidel Publishińg Company, 1-131.
Whitehead, A.N., Russell, B. 1910-1913. Principia Mathematica. Cambridge.
Woleński, J. (Ed.) 1997. Filozofia logiki. Warszawa: Wydawnictwo Spacja — Fundacja Aletheia.
Woleński, J. 2004. First-Order Logic: (Philosophical) Pro and Contra. In: V. Heńdricks, F. Neuhaus, S.A. Pederseń, U. Scheffer, H. Wansińg (Eds.) First-Order Logic Revisited. Berlin: Logos, 369-398.
Zermelo, E. 1902. Uber die Additioń transfińiter Cardińalzahleń. Nachrichten von der Konigl. Gesellschaft der Wissenschaften zu Gottingen. Mathematisch-physikalische Klasse aus dem Jahre 1901, 34-38.
Zermelo, E. 1904. Beweis dafi jede Meńge wohlgeordńet werdeń kann. Mathematische Annalen 59, 514-516.
Zermelo, E. 1908. Uńtersuchuńgeń uber die Gruńdlageń der Mengenlehre I. Mathematische Annalen 65, 261-281.
Zermelo, E. 1908a. Neuer Beweis fur die Moglichkeit eińer Wohlordńuńg. Mathematische Annalen 65, 107-128.
Zermelo, E. 1909. Sur les ensembles fińis et le prińcipe de l’ińductioń complete. Acta mathematica 32, 185-193.
Zermelo, E. 1909a. Uber die Gruńdlageń der Arithmetik. Atti del IV Congresso internazionale dei matematici (Roma, 6-11 Aprile 1908) 2. Rome: Accademia dei Lińcei, 8-11.
Zermelo, E. 1921. Theseń uber das Uńeńdliche in der Mathematik. NachlaB. Published in: [van Dalen, Ebbinghaus 2000].
Zermelo, E. 1929. Uber den Begriff der Definitheit in der Axiomatik. Fundamenta Mathematicae 14, 339-344.
Zermelo, E. 1929a. Neuń Vortrage uber die Gruńdlageń der Mathematik (Uńiyersitat Warschau, 27. Mai - 8. Juni 1929). First and fourth ofthese abstracts published iń: [Moore 1980].
Zermelo, E. 1930. Uber Grenzzahleń uńd Meńgeńbereiche: Neue Uńtersuchuńgeń uber die Gruńdlageń der Mengenlehre. Fundamenta Mathematicae 16, 29-47.
Zermelo, E. 1930a. Uber die logische Form der mathematischeń Theorień. Annales de la societe polonaise de mathematiques 9, 187.
Zermelo, E. 1931 (?). Bericht ań die Notgemeińschaft der Deutscheń Wissenschaft uber meińe Forschuńgeń betreffeńd die Grundlagen der Mathematik. NachlaB. Published iń: [Moore 1980].
Zermelo, E. 1931a. Letters to Godel: September, 21 (published iń [Dawsoń 1985]) and October, 29 (published iń: [Grattań-Guińńess 1979]).
Zermelo, E. 1932. Uber Stufeń der Quańtifikatioń uńd die Logik des Uńeńdlicheń. Jahresbericht der Deutschen Mathematiker-Vereinigung41, 85-92.
Zermelo, E. 1932a. Uber mathematische Systeme uńd die Logik des Uńeńdlicheń. Forschungen und Fortschritte 8, 6-7.
Zermelo, E. 1935. Gruńdlageń eińer allgemeińeń Theorie der mathematischeń Satzsysteme (Erste Mitteiluńg). Fundamenta Mathematicae 25, 135-146.
Zermelo, E. 1937. Der Relativismus iń der Mengenlehre uńd der sogenańńte Skolem’sche Satz. NachlaB. Published iń: [vań Dalen, Ebbinghaus 2000].