Mathematical logic (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Mathematical logic" in English language version.

Last modified: 20:06:52, 28 September 2026 (UTC)

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mathscinet.ams.org

In the foreword to the 1934 first edition of "Grundlagen der Mathematik" (Hilbert & Bernays 1934), Bernays wrote the following, which is reminiscent of the famous note by Frege when informed of Russell's paradox.
"Die Ausführung dieses Vorhabens hat eine wesentliche Verzögerung dadurch erfahren, daß in einem Stadium, in dem die Darstellung schon ihrem Abschuß nahe war, durch das Erscheinen der Arbeiten von Herbrand und von Gödel eine veränderte Situation im Gebiet der Beweistheorie entstand, welche die Berücksichtigung neuer Einsichten zur Aufgabe machte. Dabei ist der Umfang des Buches angewachsen, so daß eine Teilung in zwei Bände angezeigt erschien."
Translation:
"Carrying out this plan [by Hilbert for an exposition on proof theory for mathematical logic] has experienced an essential delay because, at the stage at which the exposition was already near to its conclusion, there occurred an altered situation in the area of proof theory due to the appearance of works by Herbrand and Gödel, which necessitated the consideration of new insights. Thus the scope of this book has grown, so that a division into two volumes seemed advisable."
So certainly Hilbert was aware of the importance of Gödel's work by 1934. The second volume in 1939 included a form of Gentzen's consistency proof for arithmetic. Hilbert, David; Bernays, Paul (1934). Grundlagen der Mathematik. I. Die Grundlehren der mathematischen Wissenschaften. Vol. 40. Berlin, New York City: Springer. ISBN 978-3-540-04134-4. JFM 60.0017.02. MR 0237246. {{cite book}}: ISBN / Date incompatibility (help)

ams.org

Woodin (2001). Woodin, W. Hugh (2001). "The Continuum Hypothesis, Part I" (PDF). Notices of the American Mathematical Society. 48 (6).

archive.org (Global: 6^th place; English: 6^th place)

Lewis Carroll: SYMBOLIC LOGIC Part I Elementary. pub. Macmillan 1896. Available online at: https://archive.org/details/symboliclogic00carr

arxiv.org (Global: 49^th place; English: 31^st place)

Hamkins & Löwe (2007). Hamkins, Joel David; Löwe, Benedikt (2007). "The modal logic of forcing". Transactions of the American Mathematical Society. 360 (4): 1793–1818. arXiv:math/0509616. doi:10.1090/s0002-9947-07-04297-3. S2CID 14724471.

doi.org (Global: 2^nd place; English: 2^nd place)

A detailed study of this terminology is given by Soare 1996. Soare, Robert I. (1996). "Computability and recursion". Bulletin of Symbolic Logic. 2 (3): 284–321. doi:10.2307/420992. JSTOR 420992. S2CID 5894394.
Ferreirós 2001 surveys the rise of first-order logic over other formal logics in the early 20th century. Ferreirós, José (2001). "The Road to Modern Logic-An Interpretation" (PDF). Bulletin of Symbolic Logic. 7 (4): 441–484. doi:10.2307/2687794. hdl:11441/38373. JSTOR 2687794. S2CID 43258676.
Ferreirós (2001), p. 443. Ferreirós, José (2001). "The Road to Modern Logic-An Interpretation" (PDF). Bulletin of Symbolic Logic. 7 (4): 441–484. doi:10.2307/2687794. hdl:11441/38373. JSTOR 2687794. S2CID 43258676.
Felscher (2000). Felscher, Walter (2000). "Bolzano, Cauchy, Epsilon, Delta". The American Mathematical Monthly. 107 (9): 844–862. doi:10.2307/2695743. JSTOR 2695743.
Zermelo (1904). Zermelo, Ernst (1904). "Beweis, daß jede Menge wohlgeordnet werden kann". Mathematische Annalen (in German). 59 (4): 514–516. doi:10.1007/BF01445300. S2CID 124189935. Archived from the original on 2016-03-05. Retrieved 2013-09-16. Reprinted in English translation as, "Proof that every set can be well-ordered" in van Heijenoort 1976, pp. 139–141.
Zermelo (1908a). Zermelo, Ernst (1908a). "Neuer Beweis für die Möglichkeit einer Wohlordnung". Mathematische Annalen (in German). 65: 107–128. doi:10.1007/BF01450054. ISSN 0025-5831. S2CID 119924143. Archived from the original on 2017-09-08. Retrieved 2013-09-16. Reprinted in English translation as, "A new proof of the possibility of a well-ordering" in van Heijenoort 1976, pp. 183–198.
Zermelo (1908b). Zermelo, Ernst (1908b). "Untersuchungen über die Grundlagen der Mengenlehre". Mathematische Annalen. 65 (2): 261–281. doi:10.1007/BF01449999. S2CID 120085563. Archived from the original on 2017-09-08. Retrieved 2013-09-16.
Ferreirós (2001), p. 445. Ferreirós, José (2001). "The Road to Modern Logic-An Interpretation" (PDF). Bulletin of Symbolic Logic. 7 (4): 441–484. doi:10.2307/2687794. hdl:11441/38373. JSTOR 2687794. S2CID 43258676.
Löwenheim (1915). Löwenheim, Leopold (1915). "Über Möglichkeiten im Relativkalkül". Mathematische Annalen (in German). 76 (4): 447–470. doi:10.1007/BF01458217. ISSN 0025-5831. S2CID 116581304. Archived from the original on 2017-09-08. Retrieved 2013-09-16. Translated as, "On possibilities in the calculus of relatives" in Jean van Heijenoort (1967). A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press. pp. 228–251.
Gentzen (1936). Gentzen, Gerhard (1936). "Die Widerspruchsfreiheit der reinen Zahlentheorie". Mathematische Annalen. 112: 132–213. doi:10.1007/BF01565428. S2CID 122719892. Reprinted in English translation in Gentzen's Collected works, M. E. Szabo, ed., North-Holland, Amsterdam, 1969.
Gödel (1958). Gödel, Kurt (1958). "Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes". Dialectica (in German). 12 (3–4): 280–287. doi:10.1111/j.1746-8361.1958.tb01464.x. Reprinted in English translation in Gödel's Collected Works, vol II, Solomon Feferman et al., eds. Oxford University Press, 1993.
Kleene (1943). Kleene, Stephen Cole (1943). "Recursive Predicates and Quantifiers". Transactions of the American Mathematical Society. 53 (1): 41–73. doi:10.2307/1990131. JSTOR 1990131.
Turing (1939). Turing, Alan M. (1939). "Systems of Logic Based on Ordinals". Proceedings of the London Mathematical Society. 45 (2): 161–228. doi:10.1112/plms/s2-45.1.161. hdl:21.11116/0000-0001-91CE-3.
Gödel (1931). Gödel, Kurt (1931). "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" [On Formally Undecidable Propositions of Principia Mathematica and Related Systems]. Monatshefte für Mathematik und Physik (in German). 38 (1): 173–198. doi:10.1007/BF01700692. S2CID 197663120.
Solovay (1976). Solovay, Robert M. (1976). "Provability Interpretations of Modal Logic". Israel Journal of Mathematics. 25 (3–4): 287–304. doi:10.1007/BF02757006. S2CID 121226261.
Hamkins & Löwe (2007). Hamkins, Joel David; Löwe, Benedikt (2007). "The modal logic of forcing". Transactions of the American Mathematical Society. 360 (4): 1793–1818. arXiv:math/0509616. doi:10.1090/s0002-9947-07-04297-3. S2CID 14724471.
Banach & Tarski (1924). Banach, Stefan; Tarski, Alfred (1924). "Sur la décomposition des ensembles de points en parties respectivement congruentes" (PDF). Fundamenta Mathematicae (in French). 6: 244–277. doi:10.4064/fm-6-1-244-277.
Morley (1965). Morley, Michael (1965). "Categoricity in Power". Transactions of the American Mathematical Society. 114 (2): 514–538. doi:10.2307/1994188. JSTOR 1994188.
Davis (1973). Davis, Martin (1973). "Hilbert's tenth problem is unsolvable". The American Mathematical Monthly. 80 (3): 233–269. doi:10.2307/2318447. JSTOR 2318447.
Reprinted as an appendix in Martin Davis (1985). Computability and Unsolvability. Dover. ISBN 978-0-486-61471-7.

enciklopedija.hr (Global: 1,079^th place; English: 4,996^th place)

"Bertić, Vatroslav". www.enciklopedija.hr. Retrieved 2023-05-01.

hdl.handle.net (Global: 66^th place; English: 34^th place)

Ferreirós 2001 surveys the rise of first-order logic over other formal logics in the early 20th century. Ferreirós, José (2001). "The Road to Modern Logic-An Interpretation" (PDF). Bulletin of Symbolic Logic. 7 (4): 441–484. doi:10.2307/2687794. hdl:11441/38373. JSTOR 2687794. S2CID 43258676.
Ferreirós (2001), p. 443. Ferreirós, José (2001). "The Road to Modern Logic-An Interpretation" (PDF). Bulletin of Symbolic Logic. 7 (4): 441–484. doi:10.2307/2687794. hdl:11441/38373. JSTOR 2687794. S2CID 43258676.
Ferreirós (2001), p. 445. Ferreirós, José (2001). "The Road to Modern Logic-An Interpretation" (PDF). Bulletin of Symbolic Logic. 7 (4): 441–484. doi:10.2307/2687794. hdl:11441/38373. JSTOR 2687794. S2CID 43258676.
Turing (1939). Turing, Alan M. (1939). "Systems of Logic Based on Ordinals". Proceedings of the London Mathematical Society. 45 (2): 161–228. doi:10.1112/plms/s2-45.1.161. hdl:21.11116/0000-0001-91CE-3.

icm.edu.pl (Global: 1,395^th place; English: 4,083^rd place)

matwbn.icm.edu.pl

Banach & Tarski (1924). Banach, Stefan; Tarski, Alfred (1924). "Sur la décomposition des ensembles de points en parties respectivement congruentes" (PDF). Fundamenta Mathematicae (in French). 6: 244–277. doi:10.4064/fm-6-1-244-277.

jaist.ac.jp (Global: low place; English: low place)

"Computability Theory and Foundations of Mathematics / February, 17th – 20th, 2014 / Tokyo Institute of Technology, Tokyo, Japan" (PDF).

jstor.org (Global: 23^rd place; English: 15^th place)

A detailed study of this terminology is given by Soare 1996. Soare, Robert I. (1996). "Computability and recursion". Bulletin of Symbolic Logic. 2 (3): 284–321. doi:10.2307/420992. JSTOR 420992. S2CID 5894394.
Ferreirós 2001 surveys the rise of first-order logic over other formal logics in the early 20th century. Ferreirós, José (2001). "The Road to Modern Logic-An Interpretation" (PDF). Bulletin of Symbolic Logic. 7 (4): 441–484. doi:10.2307/2687794. hdl:11441/38373. JSTOR 2687794. S2CID 43258676.
Ferreirós (2001), p. 443. Ferreirós, José (2001). "The Road to Modern Logic-An Interpretation" (PDF). Bulletin of Symbolic Logic. 7 (4): 441–484. doi:10.2307/2687794. hdl:11441/38373. JSTOR 2687794. S2CID 43258676.
Felscher (2000). Felscher, Walter (2000). "Bolzano, Cauchy, Epsilon, Delta". The American Mathematical Monthly. 107 (9): 844–862. doi:10.2307/2695743. JSTOR 2695743.
Ferreirós (2001), p. 445. Ferreirós, José (2001). "The Road to Modern Logic-An Interpretation" (PDF). Bulletin of Symbolic Logic. 7 (4): 441–484. doi:10.2307/2687794. hdl:11441/38373. JSTOR 2687794. S2CID 43258676.
Kleene (1943). Kleene, Stephen Cole (1943). "Recursive Predicates and Quantifiers". Transactions of the American Mathematical Society. 53 (1): 41–73. doi:10.2307/1990131. JSTOR 1990131.
Morley (1965). Morley, Michael (1965). "Categoricity in Power". Transactions of the American Mathematical Society. 114 (2): 514–538. doi:10.2307/1994188. JSTOR 1994188.
Davis (1973). Davis, Martin (1973). "Hilbert's tenth problem is unsolvable". The American Mathematical Monthly. 80 (3): 233–269. doi:10.2307/2318447. JSTOR 2318447.
Reprinted as an appendix in Martin Davis (1985). Computability and Unsolvability. Dover. ISBN 978-0-486-61471-7.

semanticscholar.org (Global: 15^th place; English: 8^th place)

api.semanticscholar.org

A detailed study of this terminology is given by Soare 1996. Soare, Robert I. (1996). "Computability and recursion". Bulletin of Symbolic Logic. 2 (3): 284–321. doi:10.2307/420992. JSTOR 420992. S2CID 5894394.
Ferreirós 2001 surveys the rise of first-order logic over other formal logics in the early 20th century. Ferreirós, José (2001). "The Road to Modern Logic-An Interpretation" (PDF). Bulletin of Symbolic Logic. 7 (4): 441–484. doi:10.2307/2687794. hdl:11441/38373. JSTOR 2687794. S2CID 43258676.
Ferreirós (2001), p. 443. Ferreirós, José (2001). "The Road to Modern Logic-An Interpretation" (PDF). Bulletin of Symbolic Logic. 7 (4): 441–484. doi:10.2307/2687794. hdl:11441/38373. JSTOR 2687794. S2CID 43258676.
Zermelo (1904). Zermelo, Ernst (1904). "Beweis, daß jede Menge wohlgeordnet werden kann". Mathematische Annalen (in German). 59 (4): 514–516. doi:10.1007/BF01445300. S2CID 124189935. Archived from the original on 2016-03-05. Retrieved 2013-09-16. Reprinted in English translation as, "Proof that every set can be well-ordered" in van Heijenoort 1976, pp. 139–141.
Zermelo (1908a). Zermelo, Ernst (1908a). "Neuer Beweis für die Möglichkeit einer Wohlordnung". Mathematische Annalen (in German). 65: 107–128. doi:10.1007/BF01450054. ISSN 0025-5831. S2CID 119924143. Archived from the original on 2017-09-08. Retrieved 2013-09-16. Reprinted in English translation as, "A new proof of the possibility of a well-ordering" in van Heijenoort 1976, pp. 183–198.
Zermelo (1908b). Zermelo, Ernst (1908b). "Untersuchungen über die Grundlagen der Mengenlehre". Mathematische Annalen. 65 (2): 261–281. doi:10.1007/BF01449999. S2CID 120085563. Archived from the original on 2017-09-08. Retrieved 2013-09-16.
Ferreirós (2001), p. 445. Ferreirós, José (2001). "The Road to Modern Logic-An Interpretation" (PDF). Bulletin of Symbolic Logic. 7 (4): 441–484. doi:10.2307/2687794. hdl:11441/38373. JSTOR 2687794. S2CID 43258676.
Löwenheim (1915). Löwenheim, Leopold (1915). "Über Möglichkeiten im Relativkalkül". Mathematische Annalen (in German). 76 (4): 447–470. doi:10.1007/BF01458217. ISSN 0025-5831. S2CID 116581304. Archived from the original on 2017-09-08. Retrieved 2013-09-16. Translated as, "On possibilities in the calculus of relatives" in Jean van Heijenoort (1967). A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press. pp. 228–251.
Gentzen (1936). Gentzen, Gerhard (1936). "Die Widerspruchsfreiheit der reinen Zahlentheorie". Mathematische Annalen. 112: 132–213. doi:10.1007/BF01565428. S2CID 122719892. Reprinted in English translation in Gentzen's Collected works, M. E. Szabo, ed., North-Holland, Amsterdam, 1969.
Gödel (1931). Gödel, Kurt (1931). "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" [On Formally Undecidable Propositions of Principia Mathematica and Related Systems]. Monatshefte für Mathematik und Physik (in German). 38 (1): 173–198. doi:10.1007/BF01700692. S2CID 197663120.
Solovay (1976). Solovay, Robert M. (1976). "Provability Interpretations of Modal Logic". Israel Journal of Mathematics. 25 (3–4): 287–304. doi:10.1007/BF02757006. S2CID 121226261.
Hamkins & Löwe (2007). Hamkins, Joel David; Löwe, Benedikt (2007). "The modal logic of forcing". Transactions of the American Mathematical Society. 360 (4): 1793–1818. arXiv:math/0509616. doi:10.1090/s0002-9947-07-04297-3. S2CID 14724471.

ucsd.edu (Global: 2,146^th place; English: 1,364^th place)

math.ucsd.edu

"Logic and Computational Complexity | Department of Mathematics". math.ucsd.edu. Retrieved 2024-12-05.

uni-goettingen.de (Global: 2,423^rd place; English: 2,569^th place)

gdz.sub.uni-goettingen.de

Zermelo (1904). Zermelo, Ernst (1904). "Beweis, daß jede Menge wohlgeordnet werden kann". Mathematische Annalen (in German). 59 (4): 514–516. doi:10.1007/BF01445300. S2CID 124189935. Archived from the original on 2016-03-05. Retrieved 2013-09-16. Reprinted in English translation as, "Proof that every set can be well-ordered" in van Heijenoort 1976, pp. 139–141.
Zermelo (1908a). Zermelo, Ernst (1908a). "Neuer Beweis für die Möglichkeit einer Wohlordnung". Mathematische Annalen (in German). 65: 107–128. doi:10.1007/BF01450054. ISSN 0025-5831. S2CID 119924143. Archived from the original on 2017-09-08. Retrieved 2013-09-16. Reprinted in English translation as, "A new proof of the possibility of a well-ordering" in van Heijenoort 1976, pp. 183–198.
Zermelo (1908b). Zermelo, Ernst (1908b). "Untersuchungen über die Grundlagen der Mengenlehre". Mathematische Annalen. 65 (2): 261–281. doi:10.1007/BF01449999. S2CID 120085563. Archived from the original on 2017-09-08. Retrieved 2013-09-16.
Löwenheim (1915). Löwenheim, Leopold (1915). "Über Möglichkeiten im Relativkalkül". Mathematische Annalen (in German). 76 (4): 447–470. doi:10.1007/BF01458217. ISSN 0025-5831. S2CID 116581304. Archived from the original on 2017-09-08. Retrieved 2013-09-16. Translated as, "On possibilities in the calculus of relatives" in Jean van Heijenoort (1967). A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press. pp. 228–251.

us.es (Global: 5,714^th place; English: 9,479^th place)

idus.us.es

Ferreirós 2001 surveys the rise of first-order logic over other formal logics in the early 20th century. Ferreirós, José (2001). "The Road to Modern Logic-An Interpretation" (PDF). Bulletin of Symbolic Logic. 7 (4): 441–484. doi:10.2307/2687794. hdl:11441/38373. JSTOR 2687794. S2CID 43258676.
Ferreirós (2001), p. 443. Ferreirós, José (2001). "The Road to Modern Logic-An Interpretation" (PDF). Bulletin of Symbolic Logic. 7 (4): 441–484. doi:10.2307/2687794. hdl:11441/38373. JSTOR 2687794. S2CID 43258676.
Ferreirós (2001), p. 445. Ferreirós, José (2001). "The Road to Modern Logic-An Interpretation" (PDF). Bulletin of Symbolic Logic. 7 (4): 441–484. doi:10.2307/2687794. hdl:11441/38373. JSTOR 2687794. S2CID 43258676.

web.archive.org (Global: 1^st place; English: 1^st place)

Zermelo (1904). Zermelo, Ernst (1904). "Beweis, daß jede Menge wohlgeordnet werden kann". Mathematische Annalen (in German). 59 (4): 514–516. doi:10.1007/BF01445300. S2CID 124189935. Archived from the original on 2016-03-05. Retrieved 2013-09-16. Reprinted in English translation as, "Proof that every set can be well-ordered" in van Heijenoort 1976, pp. 139–141.
Zermelo (1908a). Zermelo, Ernst (1908a). "Neuer Beweis für die Möglichkeit einer Wohlordnung". Mathematische Annalen (in German). 65: 107–128. doi:10.1007/BF01450054. ISSN 0025-5831. S2CID 119924143. Archived from the original on 2017-09-08. Retrieved 2013-09-16. Reprinted in English translation as, "A new proof of the possibility of a well-ordering" in van Heijenoort 1976, pp. 183–198.
Zermelo (1908b). Zermelo, Ernst (1908b). "Untersuchungen über die Grundlagen der Mengenlehre". Mathematische Annalen. 65 (2): 261–281. doi:10.1007/BF01449999. S2CID 120085563. Archived from the original on 2017-09-08. Retrieved 2013-09-16.
Löwenheim (1915). Löwenheim, Leopold (1915). "Über Möglichkeiten im Relativkalkül". Mathematische Annalen (in German). 76 (4): 447–470. doi:10.1007/BF01458217. ISSN 0025-5831. S2CID 116581304. Archived from the original on 2017-09-08. Retrieved 2013-09-16. Translated as, "On possibilities in the calculus of relatives" in Jean van Heijenoort (1967). A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press. pp. 228–251.

worldcat.org (Global: 4^th place; English: 4^th place)

search.worldcat.org

Zermelo (1908a). Zermelo, Ernst (1908a). "Neuer Beweis für die Möglichkeit einer Wohlordnung". Mathematische Annalen (in German). 65: 107–128. doi:10.1007/BF01450054. ISSN 0025-5831. S2CID 119924143. Archived from the original on 2017-09-08. Retrieved 2013-09-16. Reprinted in English translation as, "A new proof of the possibility of a well-ordering" in van Heijenoort 1976, pp. 183–198.
Löwenheim (1915). Löwenheim, Leopold (1915). "Über Möglichkeiten im Relativkalkül". Mathematische Annalen (in German). 76 (4): 447–470. doi:10.1007/BF01458217. ISSN 0025-5831. S2CID 116581304. Archived from the original on 2017-09-08. Retrieved 2013-09-16. Translated as, "On possibilities in the calculus of relatives" in Jean van Heijenoort (1967). A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press. pp. 228–251.

zbmath.org (Global: 2,089^th place; English: 1,141^st place)

In the foreword to the 1934 first edition of "Grundlagen der Mathematik" (Hilbert & Bernays 1934), Bernays wrote the following, which is reminiscent of the famous note by Frege when informed of Russell's paradox.
"Die Ausführung dieses Vorhabens hat eine wesentliche Verzögerung dadurch erfahren, daß in einem Stadium, in dem die Darstellung schon ihrem Abschuß nahe war, durch das Erscheinen der Arbeiten von Herbrand und von Gödel eine veränderte Situation im Gebiet der Beweistheorie entstand, welche die Berücksichtigung neuer Einsichten zur Aufgabe machte. Dabei ist der Umfang des Buches angewachsen, so daß eine Teilung in zwei Bände angezeigt erschien."
Translation:
"Carrying out this plan [by Hilbert for an exposition on proof theory for mathematical logic] has experienced an essential delay because, at the stage at which the exposition was already near to its conclusion, there occurred an altered situation in the area of proof theory due to the appearance of works by Herbrand and Gödel, which necessitated the consideration of new insights. Thus the scope of this book has grown, so that a division into two volumes seemed advisable."
So certainly Hilbert was aware of the importance of Gödel's work by 1934. The second volume in 1939 included a form of Gentzen's consistency proof for arithmetic. Hilbert, David; Bernays, Paul (1934). Grundlagen der Mathematik. I. Die Grundlehren der mathematischen Wissenschaften. Vol. 40. Berlin, New York City: Springer. ISBN 978-3-540-04134-4. JFM 60.0017.02. MR 0237246. {{cite book}}: ISBN / Date incompatibility (help)

Mathematical logic (English Wikipedia)

ams.org (Global: 434th place; English: 249th place)

mathscinet.ams.org

ams.org

archive.org (Global: 6th place; English: 6th place)

arxiv.org (Global: 49th place; English: 31st place)

doi.org (Global: 2nd place; English: 2nd place)

enciklopedija.hr (Global: 1,079th place; English: 4,996th place)

hdl.handle.net (Global: 66th place; English: 34th place)

icm.edu.pl (Global: 1,395th place; English: 4,083rd place)