Finite set (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Finite set" in English language version.

Last modified: 23:14:25, 25 August 2026 (UTC)

Ref.Un. Ref.Website

Global rank English rank

66doi.org

2^nd place

44icm.edu.pl

1,395^th place

4,083^rd place

44web.archive.org

1^st place

11artofproblemsolving.com

low place

11springer.com

182^nd place

172^nd place

11archive.org

6^th place

11umkc.edu

3,600^th place

2,168^th place

11ams.org

434^th place

249^th place

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

mathscinet.ams.org

de la Cruz, Dzhafarov & Hall (2006, p. 8) de la Cruz, Omar; Dzhafarov, Damir D.; Hall, Eric J. (2006), "Definitions of finiteness based on order properties" (PDF), Fundamenta Mathematicae, 189 (2): 155–172, doi:10.4064/fm189-2-5, MR 2214576

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

This list of 8 finiteness concepts is presented with this numbering scheme by both Howard & Rubin 1998, pp. 278–280, and Lévy 1958, pp. 2–3, although the details of the presentation of the definitions differ in some respects which do not affect the meanings of the concepts. Howard, Paul; Rubin, Jean E. (1998), Consequences of the axiom of choice, Providence, Rhode Island: American Mathematical Society, ISBN 9780821809778 Lévy, Azriel (1958), "The independence of various definitions of finiteness" (PDF), Fundamenta Mathematicae, 46: 1–13, doi:10.4064/fm-46-1-1-13, archived (PDF) from the original on 2003-07-05

artofproblemsolving.com (Global: low place; English: low place)

"Art of Problem Solving", artofproblemsolving.com, retrieved 2022-09-07

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

The equivalence of the standard numerical definition of finite sets to the Dedekind-finiteness of the power set of the power set was shown in 1912 by Whitehead & Russell 2009, p. 288. This Whitehead/Russell theorem is described in more modern language by Tarski 1924, pp. 73–74. Whitehead, Alfred North; Russell, Bertrand (February 2009) [1912], Principia Mathematica, vol. Two, Merchant Books, ISBN 978-1-60386-183-0 Tarski, Alfred (1924), "Sur les ensembles finis" (PDF), Fundamenta Mathematicae, 6: 45–95, doi:10.4064/fm-6-1-45-95, archived (PDF) from the original on 2011-05-15
Tarski 1924, pp. 48–58, demonstrated that his definition (which is also known as I-finite) is equivalent to Kuratowski's set-theoretical definition, which he then noted is equivalent to the standard numerical definition via the proof by Kuratowski 1920, pp. 130–131. Tarski, Alfred (1924), "Sur les ensembles finis" (PDF), Fundamenta Mathematicae, 6: 45–95, doi:10.4064/fm-6-1-45-95, archived (PDF) from the original on 2011-05-15 Kuratowski, Kazimierz (1920), "Sur la notion d'ensemble fini" (PDF), Fundamenta Mathematicae, 1: 129–131, doi:10.4064/fm-1-1-129-131, archived (PDF) from the original on 2011-05-15
Herrlich, Horst (2006), "Proposition 4.13", Axiom of Choice, Lecture Notes in Mathematics, vol. 1876, Springer, p. 48, doi:10.1007/11601562, ISBN 3-540-30989-6, retrieved 18 July 2023
This list of 8 finiteness concepts is presented with this numbering scheme by both Howard & Rubin 1998, pp. 278–280, and Lévy 1958, pp. 2–3, although the details of the presentation of the definitions differ in some respects which do not affect the meanings of the concepts. Howard, Paul; Rubin, Jean E. (1998), Consequences of the axiom of choice, Providence, Rhode Island: American Mathematical Society, ISBN 9780821809778 Lévy, Azriel (1958), "The independence of various definitions of finiteness" (PDF), Fundamenta Mathematicae, 46: 1–13, doi:10.4064/fm-46-1-1-13, archived (PDF) from the original on 2003-07-05
de la Cruz, Dzhafarov & Hall (2006, p. 8) de la Cruz, Omar; Dzhafarov, Damir D.; Hall, Eric J. (2006), "Definitions of finiteness based on order properties" (PDF), Fundamenta Mathematicae, 189 (2): 155–172, doi:10.4064/fm189-2-5, MR 2214576
Lévy 1958 found counter-examples to each of the reverse implications in Mostowski models. Lévy attributes most of the results to earlier papers by Mostowski and Lindenbaum. Lévy, Azriel (1958), "The independence of various definitions of finiteness" (PDF), Fundamenta Mathematicae, 46: 1–13, doi:10.4064/fm-46-1-1-13, archived (PDF) from the original on 2003-07-05

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

matwbn.icm.edu.pl

The equivalence of the standard numerical definition of finite sets to the Dedekind-finiteness of the power set of the power set was shown in 1912 by Whitehead & Russell 2009, p. 288. This Whitehead/Russell theorem is described in more modern language by Tarski 1924, pp. 73–74. Whitehead, Alfred North; Russell, Bertrand (February 2009) [1912], Principia Mathematica, vol. Two, Merchant Books, ISBN 978-1-60386-183-0 Tarski, Alfred (1924), "Sur les ensembles finis" (PDF), Fundamenta Mathematicae, 6: 45–95, doi:10.4064/fm-6-1-45-95, archived (PDF) from the original on 2011-05-15
Tarski 1924, pp. 48–58, demonstrated that his definition (which is also known as I-finite) is equivalent to Kuratowski's set-theoretical definition, which he then noted is equivalent to the standard numerical definition via the proof by Kuratowski 1920, pp. 130–131. Tarski, Alfred (1924), "Sur les ensembles finis" (PDF), Fundamenta Mathematicae, 6: 45–95, doi:10.4064/fm-6-1-45-95, archived (PDF) from the original on 2011-05-15 Kuratowski, Kazimierz (1920), "Sur la notion d'ensemble fini" (PDF), Fundamenta Mathematicae, 1: 129–131, doi:10.4064/fm-1-1-129-131, archived (PDF) from the original on 2011-05-15
This list of 8 finiteness concepts is presented with this numbering scheme by both Howard & Rubin 1998, pp. 278–280, and Lévy 1958, pp. 2–3, although the details of the presentation of the definitions differ in some respects which do not affect the meanings of the concepts. Howard, Paul; Rubin, Jean E. (1998), Consequences of the axiom of choice, Providence, Rhode Island: American Mathematical Society, ISBN 9780821809778 Lévy, Azriel (1958), "The independence of various definitions of finiteness" (PDF), Fundamenta Mathematicae, 46: 1–13, doi:10.4064/fm-46-1-1-13, archived (PDF) from the original on 2003-07-05
Lévy 1958 found counter-examples to each of the reverse implications in Mostowski models. Lévy attributes most of the results to earlier papers by Mostowski and Lindenbaum. Lévy, Azriel (1958), "The independence of various definitions of finiteness" (PDF), Fundamenta Mathematicae, 46: 1–13, doi:10.4064/fm-46-1-1-13, archived (PDF) from the original on 2003-07-05

springer.com (Global: 182^nd place; English: 172^nd place)

link.springer.com

Herrlich, Horst (2006), "Proposition 4.13", Axiom of Choice, Lecture Notes in Mathematics, vol. 1876, Springer, p. 48, doi:10.1007/11601562, ISBN 3-540-30989-6, retrieved 18 July 2023

umkc.edu (Global: 3,600^th place; English: 2,168^th place)

h.web.umkc.edu

de la Cruz, Dzhafarov & Hall (2006, p. 8) de la Cruz, Omar; Dzhafarov, Damir D.; Hall, Eric J. (2006), "Definitions of finiteness based on order properties" (PDF), Fundamenta Mathematicae, 189 (2): 155–172, doi:10.4064/fm189-2-5, MR 2214576

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

The equivalence of the standard numerical definition of finite sets to the Dedekind-finiteness of the power set of the power set was shown in 1912 by Whitehead & Russell 2009, p. 288. This Whitehead/Russell theorem is described in more modern language by Tarski 1924, pp. 73–74. Whitehead, Alfred North; Russell, Bertrand (February 2009) [1912], Principia Mathematica, vol. Two, Merchant Books, ISBN 978-1-60386-183-0 Tarski, Alfred (1924), "Sur les ensembles finis" (PDF), Fundamenta Mathematicae, 6: 45–95, doi:10.4064/fm-6-1-45-95, archived (PDF) from the original on 2011-05-15
Tarski 1924, pp. 48–58, demonstrated that his definition (which is also known as I-finite) is equivalent to Kuratowski's set-theoretical definition, which he then noted is equivalent to the standard numerical definition via the proof by Kuratowski 1920, pp. 130–131. Tarski, Alfred (1924), "Sur les ensembles finis" (PDF), Fundamenta Mathematicae, 6: 45–95, doi:10.4064/fm-6-1-45-95, archived (PDF) from the original on 2011-05-15 Kuratowski, Kazimierz (1920), "Sur la notion d'ensemble fini" (PDF), Fundamenta Mathematicae, 1: 129–131, doi:10.4064/fm-1-1-129-131, archived (PDF) from the original on 2011-05-15
This list of 8 finiteness concepts is presented with this numbering scheme by both Howard & Rubin 1998, pp. 278–280, and Lévy 1958, pp. 2–3, although the details of the presentation of the definitions differ in some respects which do not affect the meanings of the concepts. Howard, Paul; Rubin, Jean E. (1998), Consequences of the axiom of choice, Providence, Rhode Island: American Mathematical Society, ISBN 9780821809778 Lévy, Azriel (1958), "The independence of various definitions of finiteness" (PDF), Fundamenta Mathematicae, 46: 1–13, doi:10.4064/fm-46-1-1-13, archived (PDF) from the original on 2003-07-05
Lévy 1958 found counter-examples to each of the reverse implications in Mostowski models. Lévy attributes most of the results to earlier papers by Mostowski and Lindenbaum. Lévy, Azriel (1958), "The independence of various definitions of finiteness" (PDF), Fundamenta Mathematicae, 46: 1–13, doi:10.4064/fm-46-1-1-13, archived (PDF) from the original on 2003-07-05

Finite set (English Wikipedia)

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

mathscinet.ams.org

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

artofproblemsolving.com (Global: low place; English: low place)

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

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