The compactness of first-order logic:from gödel to lindström

References

• Barwise , J. “ An introduction to first-order logic ” . In Handbook of mathematical logic Edited by: Barwise , J. 5 – 46 .
   Google Scholar
• Bernays , P. 1926 . Axiomatische Untersuchung des Aussagen-Kalküls der Principia Mathematica . Mathematische Zeitschrift , 25 : 305 – 320 .
   Google Scholar
• Beth , E.W. 1954 . A topological proof of the theorem of Löwenheim-Skolem-Gödel . Indagationes Mathematical , 13 : 436 – 444 .
   Google Scholar
• Cartan , H. 1937a . Théorie des filtres . Comptes rendus hebdomadaires des séances de l’Académie des Sciences , 205 : 595 – 598 .
   Google Scholar
• Cartan , H. 1937b . Filtres et ultrafiltres . Comptes rendus hebdomadaires des séances de VAcadémie des Sciences , 205 : 777 – 779 .
   Google Scholar
• Chang , C.C. and Keisler , H.J. 1973 . “ Model theory ” . In Studies in Logic and the Foundations of Mathematics , Vol. 73 , Amsterdam : North-Holland .
   Google Scholar
• Church , A. 1956 . “ Introduction to mathematical logic ” . Vol. 1 , Princeton : University Press .
   Google Scholar
• Corcoran , J. 1969 . Three logical theories . Philosophy of science , 36 : 153 – 177 .
   Web of Science ®Google Scholar
• Curry , H. 1963 . Foundation of mathematical logic , New York : McGraw-Hill .
   Google Scholar
• Ebbinghaus , H.D. 1985 . “ Extended logics:The general framework ” . In Model-theoretic logics , Edited by: Barwise , J. and Feferman , S. 25 – 76 . New York : Springer .
   Google Scholar
• Eklof , P.C. 1977 . “ Ultraproducts for algebraists ” . In Handbook of mathematical logic , Edited by: Barwise , J. 105 – 137 . Amsterdam : North-Holland .
   Google Scholar
• Feferman , S. and Tarski , A. 1953 . Journal of Symbolic Logic , 18 : 339 – 340 . Review of Rasiowa and Sikorski 1952
   Google Scholar
• Flum , J. 1985 . “ Characterizing logics ” . In Model-theoretic logics , Edited by: Barwise , J. and Feferman , S. 77 – 120 . New York : Springer .
   Google Scholar
• Frayne , T. , Morel , A.C. and Scott , D.S. 1962 . Reduced direct products . Fundamenta mathematicae , 51 : 195 – 228 .
   Google Scholar
• Fuhrken , E.G. 1964 . Skolem-type normal forms for first-order languages with a generalized quantifier . Fundamenta mathematicae , 54 : 291 – 302 .
   Google Scholar
• Gödel , K. 1929 . Über die Vollständigkeit des Logikkalküls , University of Vienna . doctoral dissertation
   Google Scholar
• Gödel , K. 1930 . Die Vollständigkeit der Axiome des logischen Funktionenkalküls . Monatshefte für Mathematik und Physik , 37 : 349 – 360 .
   Google Scholar
• Gödel , K. 1932 . Eine Eigenschaft der Realisierungen des Aussagenkalküls . Ergebnisse eines mathematischen Kolloquiums , 3 : 20 – 21 .
   Google Scholar
• Gödel , K. 1934a . Zentralblatt für Mathematik und ihre Grenzgebiete , 7 : 97 – 98 . Review of Skolem 1934
   Google Scholar
• Gödel , K. 1934b . “ On undecidable propositions of formal mathematical systems ” . 346 – 371 . Gödel . 1986
   Google Scholar
• Gödel , K. 1986 . Collected Works , Vol. I , New York : Oxford University Press .
   Google Scholar
• Henkin , L. 1947 . The completeness of formal systems , Princeton University . dissertation
   Google Scholar
• Henkin , L. 1949a . Fragments of the propositional calculus . Journal of symbolic logic , 14 : 42 – 48 .
   Google Scholar
• Henkin , L. 1949b . The completeness of the first-order functional calculus . Journal of Symbolic Logic , 14 : 159 – 166 .
   Google Scholar
• Henkin , L. 1950 . Completeness in the theory of types . Journal of symbolic logic , 15 : 81 – 91 .
   Google Scholar
• Henkin , L. 1952 . Journal of symbolic logic , 17 : 205 – 207 . Review of Robinson 1951
   Google Scholar
• Henkin , L. 1953 . Some interconnections between modern algebra and mathematical logic . Transactions of the American Mathematical Society , 74 : 410 – 427 .
   Web of Science ®Google Scholar
• Henkin , L. 1954 . Metamathematical theorems equivalent to the Prime Ideal Theorem for Boolean algebra . Bulletin of the American Mathematical Society , 60 : 387 – 388 .
   Web of Science ®Google Scholar
• Henkin , L. 1955 . Journal of symbolic logic , 20 : 185 – 186 . Review of Robinson 1953
   Google Scholar
• Henkin , L. and Mostowski , A. 1959 . Journal of symbolic logic , 24 : 55 – 57 . Review of Maltsev 1941 and 1956
   Google Scholar
• Hewitt , E. 1948 . Rings of real-valued continuous functions . Transactions of the American Mathematical Society , 64 : 45 – 99 .
   Web of Science ®Google Scholar
• Howard , P. 1990 . Definitions of compact . Journal of symbolic logic , 55 : 645 – 655 .
   Web of Science ®Google Scholar
• Kaufmann , M. 1985 . “ The quantifier "There exist uncountably many" and some of its relatives ” . In Model-theoretic logics , Edited by: Barwise , J. and Feferman , S. 123 – 176 . New York : Springer .
   Google Scholar
• Keisler , H.J. 1965 . “ A survey of ultraproducts ” . In Logic, methodology, and philosophy of science , Edited by: Bar-Hillel , Y. 112 – 126 . Amsterdam : North-Holland .
   Google Scholar
• Keisler , H.J. 1977 . “ Fundamentals of model theory ” . In Handbook of mathematical logic , Edited by: Barwise , J. 47 – 103 . Amsterdam : North-Holland .
   Google Scholar
• Kleene , S.C. 1952 . Introduction to metamathematics , Amsterdam : North-Holland .
   Google Scholar
• Kreisel , G. and Krivine , J.L. 1971 . Elements of mathematical logic model theory , Amsterdam : North-Holland .
   Google Scholar
• Lindström , P. 1969 . On extensions of elementary logic . Theoria , 35 : 3 – 11 .
   Google Scholar
• Loś , J. 1955 . “ Quelques remarques, théorèmes et problèmes sur les classes définissables d’algèbres ” . In Mathematical interpretation of formal systems , Edited by: Brouwer , L.E.J. , Beth , E.W. and Heyting , A. 98 – 113 . Amsterdam : North-Holland . Studies in logic and the foundations of mathematics
   Google Scholar
• Löwenheim , L. 1915 . Über Möglichkeiten im Relativkalkül . Mathematische Annalen , 76 : 447 – 470 .
   Google Scholar
• Maltsev , A.I. 1936 . Untersuchungen aus dem Gebiete der mathematischen Logik . Matematicheskii Sbornik , 1 : 323 – 336 .
   Google Scholar
• Maltsev , A.I. 1940 . On the faithful representation of infinite groups by matrices . Matematicheskii Sbornik , 8 ( 50 ) : 405 – 422 . In Russian
   Google Scholar
• Maltsev, A.I., (1941) "Ob odnom obscem metode polucenia lokal’nyh teorem grupp; Ivanovskii Gosudarstvennyi Pedagogiceskii Institut, Ucenye zapiski, Fiziko-matematicheskie nauki, 1, no. 1, 3-9
   Google Scholar
• Maltsev , A.I. 1956 . O predstavleniah modelej , Vol. 108 , 27 – 29 . SSSR : Doklady Akadmeii Nauk .
   Google Scholar
• Maltsev , A.I. 1971 . The metamathematics of algebraic systems , Edited by: Wells , B.F. III . Amsterdam : North-Holland .
   Google Scholar
• Mendelson , E. 1964 . “ Introduction to mathematical logic ” . Princeton : Van Nostrand .
   Google Scholar
• Monk , J.D. 1976 . Mathematical logic , New York : Springer .
   Google Scholar
• Moore , G.H. 1982 . “ Zermelo’s axiom of choice, Its origins, development, and influence ” . In Studies in the History of Mathematics and Physical Sciences , Vol. 8 , New York : Springer .
   Google Scholar
• Moore , G.H. 1988 . “ The emergence of first-order logic ” . In History and philosophy of modern mathematics , Edited by: Aspray , W. and Kitcher , P. Vol. 11 , 95 – 135 . Minneapolis : University of Minnesota Press . Minnesota Studies in the Philosophy of Science
   Google Scholar
• Mostowski , A. 1937 . Abzählbare Boolesche Körper und ihre Anwendung auf die allgemeine Metamathematik . Fundamenta mathematicae , 29 : 34 – 53 .
   Google Scholar
• Mostowski , A. Sur l’interprétation géométrique et topologique des notions logiques . Proceedings of the Tenth International Congress of Philosophy . Amsterdam. pp. 767 – 769 . North-Holland .
   Google Scholar
• Mostowski , A. 1957 . On a generalization of quantifiers . Fundamenta mathematicae , 44 : 12 – 36 .
   Google Scholar
• Neumann , B.H. 1954 . An embedding theorem for algebraic systems . Proceedings of the London Mathematical Society , 4 ( 2 ) : 138 – 153 .
   Google Scholar
• Pier , J.P. 1980 . Historique de la notion de compacité . Historia mathematica , 7 ( 2 ) : 425 – 443 .
   Google Scholar
• Post , E.L. 1921 . Introduction to a general theory of elementary propositions . American journal of mathematics , 43 ( 2 ) : 169 – 173 .
   Google Scholar
• Rasiowa , H. 1952 . A proof of the compactness theorem for arithmetical classes . Fundamenta mathematicae , 39 ( 2 ) : 8 – 14 .
   Google Scholar
• Rasiowa , H. and Sikorski , R. 1950 . A proof of the compactness theorem of Gödel . Fundamenta mathematicae , 37 ( 2 ) : 193 – 200 .
   Google Scholar
• Robinson , A. 1949 . Journal of symbolic logic , 14 ( 2 ) : 74 Abstract of paper related to 7957
   Google Scholar
• Robinson , A. 1951 . “ On the metamathematics of algebra ” . Amsterdam : North-Holland .
   Google Scholar
• Robinson , A. Application of symbolic logic to algebra . Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, U.S.A. August30-September 6, 1950 . pp. 686 – 694 . Providence, R.I American Mathematical Society
   Google Scholar
• Robinson , A. 1953 . Les rapports entre le calcul dèductif et l’interprétation sémantique d’un système axiomatique . Colloques internationaux du Centre National de la Recherche Scientifique , 36 : 35 – 51 .
   Google Scholar
• Robinson , A. 1955 . Note on an embedding theorem for algebraic systems . Journal of the London Mathematical Society , 30 ( 2 ) : 249 – 252 .
   Google Scholar
• Robinson , A. Recent develpoment in model theory . Proceedings of the 1960 International Congress for Logic, Methodology, and Philosophy of Science . Edited by: Nagel , E. , Suppes , P. and Tarski , A. Vol. 1 , pp. 705 – 720 . Stanford : Stanford University Press .
   Google Scholar
• Robinson , A. 1979 . “ Selected Papers ” . In Model theory and algebra , Vol. I , New Haven, Conn : Yale University Press .
   Google Scholar
• Rosser , J.B. 1937 . Journal of symbolic logic , 2 : 84
   Google Scholar
• Sabbagh , G. 1991 . Bulletin of the American Mathematical Society , 25 : 215 – 221 .
   Google Scholar
• Scott , D. On constructing models of arithmetic . Infinitistic Methods Proceedings of the Symposium on the Foundations of Mathematics . 1959 , Warszawa. pp. 235 – 255 . Warsaw (PWN) .
   Google Scholar
• Seligman , G.B. 1979 . Biography of Abraham Robinson , Robinson . 1979
   Google Scholar
• Skolem , T. 1920 . Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen . Skrifter utgit av Videnskapssel slapet i Kristiania, I. Matematisk-naturvidenskabelig klasse , 6 : 1 – 38 .
   Google Scholar
• Skolem, T.1923 Einige Bemerkungen zur axiomatschen Begründung der Mengenlehre in Matematikerkongressen in Helsingfors 4-7 Juli 1922, Den femte skandinaviska matematikerkongressen, Redogoerelse, Helsinki (Akademiska Bokhandlen), 217-232
   Google Scholar
• Skolem , T. 1929 . Über einige Grundlagenfragen der Mathematik", Skrifter utgitt av Det Norske Videnskaps-Akademi i Oslo . I. Matematisk-naturvidenskapelig klasse , 7 : 1 – 49 .
   Google Scholar
• Skolem , T. 1934 . Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen . Fundamenta mathematicae , 23 : 150 – 161 .
   Google Scholar
• Skolem , T. 1941 . “ Sur la portée du théorème de Löwenheim-Skolem ” . In Les entretiens de Zurich sur les fondements et la méthode des sciences mathématiques , Edited by: Gonseth , F. 6 – 9 . Zurich : Leemann . 6-9 décembre 1938
   Google Scholar
• Stone , M.H. Boolean algebras and their application to topology . Proceedings of the National Academy of Science . USA. Vol. 20 , pp. 197 – 202 .
   Google Scholar
• Stone , M.H. 1936 . The theory of representations for Boolean algebras . Transactions of the American Mathematical Society , 40 : 37 – 111 .
   Google Scholar
• Stone , M.H. 1937 . Applications of the theory of Boolean rings to general topology . Transactions of the American Mathematical Society , 41 : 375 – 481 .
   Google Scholar
• Tarski , A. 1930 . Über einige fundamentale Begriffe der Metamathematik . Sprawozdania z posiedzen Towarzystwa Naukowego Warszawskiego, wydzial III , 23 : 22 – 29 .
   Google Scholar
• Tarski , A. 1931 . Sur les ensembles définissables de nombres réels. I . Fundamenta mathematicae , 17 : 210 – 239 .
   Google Scholar
• Tarski , A. 1936 . Grundzüge des Systemenkalküls, Zweiter Teil . Fundamenta mathematicae , 26 : 283 – 301 .
   Google Scholar
• Tarski , A. Some notions and methods on the borderline of algebra and metamathematics . Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950 . Vol. 1 , pp. 705 – 720 . Providence, R.I : American Mathematical Society .
   Google Scholar
• Van Heijenoort , J. , ed. 1967 . From Frege to Gödel:a source book in mathematical logic , 1879 – 1931 . Cambridge, Mass : Harvard University Press .
   Google Scholar
• Van Heijenoort, J., and Dreben, B., 1986 [introductory note to Gödel 1929, 1930, and 1930a], in Gödel 1986, 44-59
   Google Scholar
• Vaught , R. Model theory before 1945 . Proceedings of the Tarski Symposium (Proceedings of Symposia in Pure Mathematics . Edited by: Henkin , L. Vol. 25 , pp. 153 – 172 . New York : American Mathematical Society . Proceedings of Symposia in Pure Mathematics
   Google Scholar
• Vaught, R., 1986 [Introductory note to Gödel 1934c and 1935], in Gödel 1986, 376-379
   Google Scholar
• Wang , H. 1974 . From mathematics to philosophy , London : Routledge and Kegan Paul .
   Google Scholar
• Zygmunt , J. 1973 . On the sources of the notion of the reduced product . Reports on mathematical logic , 1 : 53 – 67 .
   Google Scholar