Aksioma Peano (Indonesian Wikipedia)
Analysis of information sources in references of the Wikipedia article "Aksioma Peano" in Indonesian language version.
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- Willard 2001. Willard, Dan E. (2001). "Self-verifying axiom systems, the incompleteness theorem and related reflection principles" (PDF). The Journal of Symbolic Logic. 66 (2): 536–596. doi:10.2307/2695030. MR 1833464. Diarsipkan dari versi asli (PDF) tanggal 2020-11-09. Diakses tanggal 2020-07-16.
ams.org
- Peirce 1881, Shields 1997 Peirce, C. S. (1881). "On the Logic of Number". American Journal of Mathematics. 4: 85–95. doi:10.2307/2369151. JSTOR 2369151. MR 1507856. Shields, Paul (1997). "3. Peirce's Axiomatization of Arithmetic". Dalam Houser, Nathan; Roberts, Don D.; Van Evra, James. Studies in the Logic of Charles Sanders Peirce
. Indiana University Press. hlm. 43–52. ISBN 0-253-33020-3. - Hilbert 1902. Hilbert, David (1902). Diterjemahkan oleh Winton, Maby. "Mathematische Probleme" [Mathematical Problems]. Bulletin of the American Mathematical Society. 8: 437–479. doi:10.1090/s0002-9904-1902-00923-3
. - Willard 2001. Willard, Dan E. (2001). "Self-verifying axiom systems, the incompleteness theorem and related reflection principles" (PDF). The Journal of Symbolic Logic. 66 (2): 536–596. doi:10.2307/2695030. MR 1833464. Diarsipkan dari versi asli (PDF) tanggal 2020-11-09. Diakses tanggal 2020-07-16.
archive.org
- Peirce 1881, Shields 1997 Peirce, C. S. (1881). "On the Logic of Number". American Journal of Mathematics. 4: 85–95. doi:10.2307/2369151. JSTOR 2369151. MR 1507856. Shields, Paul (1997). "3. Peirce's Axiomatization of Arithmetic". Dalam Houser, Nathan; Roberts, Don D.; Van Evra, James. Studies in the Logic of Charles Sanders Peirce. Indiana University Press. hlm. 43–52. ISBN 0-253-33020-3.
- van Heijenoort 1967, hlm. 94. van Heijenoort, Jean (1967). From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press. ISBN 9780674324497.
- Berisi terjemahan dari dua makalah berikut, dilengkapi komentar:
- Dedekind, Richard (1890). Letter to Keferstein. On p. 100, he restates and defends his axioms of 1888. hlm. 98–103.
- van Heijenoort 1967, hlm. 2. van Heijenoort, Jean (1967). From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press. ISBN 9780674324497.
- Berisi terjemahan dari dua makalah berikut, dilengkapi komentar:
- Dedekind, Richard (1890). Letter to Keferstein. On p. 100, he restates and defends his axioms of 1888. hlm. 98–103.
- van Heijenoort 1967, hlm. 83. van Heijenoort, Jean (1967). From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press. ISBN 9780674324497.
- Berisi terjemahan dari dua makalah berikut, dilengkapi komentar:
- Dedekind, Richard (1890). Letter to Keferstein. On p. 100, he restates and defends his axioms of 1888. hlm. 98–103.
- Peano 1889, hlm. 1. Peano, Giuseppe (1889). Arithmetices principia, nova methodo exposita [The principles of arithmetic, presented by a new method]. An excerpt of the treatise where Peano first presented his axioms, and recursively defined arithmetical operations. hlm. 83–97.
- Fritz 1952, p. 137
An illustration of 'interpretation' is Russell's own definition of 'cardinal number'. The uninterpreted system in this case is Peano's axioms for the number system, whose three primitive ideas and five axioms, Peano believed, were sufficient to enable one to derive all the properties of the system of natural numbers. Actually, Russell maintains, Peano's axioms define any progression of the form of which the series of the natural numbers is one instance. Fritz, Charles A., Jr. (1952). Bertrand Russell's construction of the external world. - Gray 2013, p. 133
So Poincaré turned to see whether logicism could generate arithmetic, more precisely, the arithmetic of ordinals. Couturat, said Poincaré, had accepted the Peano axioms as a definition of a number. But this will not do. The axioms cannot be shown to be free of contradiction by finding examples of them, and any attempt to show that they were contradiction-free by examining the totality of their implications would require the very principle of mathematical induction Couturat believed they implied. For (in a further passage dropped from S&M) either one assumed the principle in order to prove it, which would only prove that if it is true it is not self-contradictory, which says nothing; or one used the principle in another form than the one stated, in which case one must show that the number of steps in one's reasoning was an integer according to the new definition, but this could not be done (1905c, 834). Gray, Jeremy (2013). "The Essayist". Henri Poincaré: A scientific biography. Princeton University Press. hlm. 133. ISBN 0-691-15271-3.
books.google.com
- Peirce 1881, Shields 1997 Peirce, C. S. (1881). "On the Logic of Number". American Journal of Mathematics. 4: 85–95. doi:10.2307/2369151. JSTOR 2369151. MR 1507856. Shields, Paul (1997). "3. Peirce's Axiomatization of Arithmetic". Dalam Houser, Nathan; Roberts, Don D.; Van Evra, James. Studies in the Logic of Charles Sanders Peirce. Indiana University Press. hlm. 43–52. ISBN 0-253-33020-3.
- Gray 2013, p. 133
So Poincaré turned to see whether logicism could generate arithmetic, more precisely, the arithmetic of ordinals. Couturat, said Poincaré, had accepted the Peano axioms as a definition of a number. But this will not do. The axioms cannot be shown to be free of contradiction by finding examples of them, and any attempt to show that they were contradiction-free by examining the totality of their implications would require the very principle of mathematical induction Couturat believed they implied. For (in a further passage dropped from S&M) either one assumed the principle in order to prove it, which would only prove that if it is true it is not self-contradictory, which says nothing; or one used the principle in another form than the one stated, in which case one must show that the number of steps in one's reasoning was an integer according to the new definition, but this could not be done (1905c, 834). Gray, Jeremy (2013). "The Essayist". Henri Poincaré: A scientific biography. Princeton University Press. hlm. 133. ISBN 0-691-15271-3.
doi.org
- Peirce 1881, Shields 1997 Peirce, C. S. (1881). "On the Logic of Number". American Journal of Mathematics. 4: 85–95. doi:10.2307/2369151. JSTOR 2369151. MR 1507856. Shields, Paul (1997). "3. Peirce's Axiomatization of Arithmetic". Dalam Houser, Nathan; Roberts, Don D.; Van Evra, James. Studies in the Logic of Charles Sanders Peirce. Indiana University Press. hlm. 43–52. ISBN 0-253-33020-3.
- Hilbert 1902. Hilbert, David (1902). Diterjemahkan oleh Winton, Maby. "Mathematische Probleme" [Mathematical Problems]. Bulletin of the American Mathematical Society. 8: 437–479. doi:10.1090/s0002-9904-1902-00923-3 .
- Gödel 1931. Gödel, Kurt (1931). See On Formally Undecidable Propositions of Principia Mathematica and Related Systems for details on English translations. "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I" (PDF). Monatshefte für Mathematik. 38: 173–198. doi:10.1007/bf01700692. Diarsipkan dari versi asli (PDF) tanggal 2018-04-11. Diakses tanggal 2013-10-31.
- Gödel 1958 Gödel, Kurt (1958). Reprinted in English translation in 1990. Gödel's Collected Works, Vol II. Solomon Feferman et al., eds. "Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes". Dialectica. Oxford University Press. 12: 280–287. doi:10.1111/j.1746-8361.1958.tb01464.x.
- Gentzen 1936 Gentzen, Gerhard (1936). Reprinted in English translation in his 1969 Collected works, M. E. Szabo, ed. "Die Widerspruchsfreiheit der reinen Zahlentheorie". Mathematische Annalen. 112: 132–213. doi:10.1007/bf01565428.
- Willard 2001. Willard, Dan E. (2001). "Self-verifying axiom systems, the incompleteness theorem and related reflection principles" (PDF). The Journal of Symbolic Logic. 66 (2): 536–596. doi:10.2307/2695030. MR 1833464. Diarsipkan dari versi asli (PDF) tanggal 2020-11-09. Diakses tanggal 2020-07-16.
google.ca
books.google.ca
- Fritz 1952, p. 137
An illustration of 'interpretation' is Russell's own definition of 'cardinal number'. The uninterpreted system in this case is Peano's axioms for the number system, whose three primitive ideas and five axioms, Peano believed, were sufficient to enable one to derive all the properties of the system of natural numbers. Actually, Russell maintains, Peano's axioms define any progression of the form of which the series of the natural numbers is one instance. Fritz, Charles A., Jr. (1952). Bertrand Russell's construction of the external world.
jstor.org
- Peirce 1881, Shields 1997 Peirce, C. S. (1881). "On the Logic of Number". American Journal of Mathematics. 4: 85–95. doi:10.2307/2369151. JSTOR 2369151. MR 1507856. Shields, Paul (1997). "3. Peirce's Axiomatization of Arithmetic". Dalam Houser, Nathan; Roberts, Don D.; Van Evra, James. Studies in the Logic of Charles Sanders Peirce. Indiana University Press. hlm. 43–52. ISBN 0-253-33020-3.
uni-potsdam.de
- Grassmann 1861. Grassmann, Hermann (1861). "Lehrbuch der Arithmetik" [A tutorial in arithmetic] (PDF). Enslin. Diarsipkan dari versi asli (PDF) tanggal 2013-11-03. Diakses tanggal 2020-06-25.
w-k-essler.de
- Gödel 1931. Gödel, Kurt (1931). See On Formally Undecidable Propositions of Principia Mathematica and Related Systems for details on English translations. "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I" (PDF). Monatshefte für Mathematik. 38: 173–198. doi:10.1007/bf01700692. Diarsipkan dari versi asli (PDF) tanggal 2018-04-11. Diakses tanggal 2013-10-31.
web.archive.org
- Grassmann 1861. Grassmann, Hermann (1861). "Lehrbuch der Arithmetik" [A tutorial in arithmetic] (PDF). Enslin. Diarsipkan dari versi asli (PDF) tanggal 2013-11-03. Diakses tanggal 2020-06-25.
- Gödel 1931. Gödel, Kurt (1931). See On Formally Undecidable Propositions of Principia Mathematica and Related Systems for details on English translations. "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I" (PDF). Monatshefte für Mathematik. 38: 173–198. doi:10.1007/bf01700692. Diarsipkan dari versi asli (PDF) tanggal 2018-04-11. Diakses tanggal 2013-10-31.
- Willard 2001. Willard, Dan E. (2001). "Self-verifying axiom systems, the incompleteness theorem and related reflection principles" (PDF). The Journal of Symbolic Logic. 66 (2): 536–596. doi:10.2307/2695030. MR 1833464. Diarsipkan dari versi asli (PDF) tanggal 2020-11-09. Diakses tanggal 2020-07-16.