Peano axioms (English Wikipedia)
Analysis of information sources in references of the Wikipedia article "Peano axioms" in English 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. JSTOR 2695030. MR 1833464. S2CID 2822314.
ams.org
mathscinet.ams.org
- Peirce 1881. Peirce, C. S. (1881). "On the Logic of Number". American Journal of Mathematics. 4 (1): 85–95. doi:10.2307/2369151. JSTOR 2369151. MR 1507856.
- 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. JSTOR 2695030. MR 1833464. S2CID 2822314.
ams.org
- Hilbert 1902. Hilbert, David (1902). "Mathematische Probleme" [Mathematical Problems]. Bulletin of the American Mathematical Society. 8 (10). Translated by Winton, Maby: 437–479. doi:10.1090/s0002-9904-1902-00923-3.
archive.org
- Peirce 1881. Peirce, C. S. (1881). "On the Logic of Number". American Journal of Mathematics. 4 (1): 85–95. doi:10.2307/2369151. JSTOR 2369151. MR 1507856.
- Shields 1997. Shields, Paul (1997). "3. Peirce's Axiomatization of Arithmetic". In Houser, Nathan; Roberts, Don D.; Van Evra, James (eds.). Studies in the Logic of Charles Sanders Peirce. Indiana University Press. pp. 43–52. ISBN 0-253-33020-3.
- Peano 1889, p. 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. Fratres Bocca. pp. 83–97.
- Peano 1908, p. 27. Peano, Giuseppe (1908). Formulario Mathematico (V ed.). Turin, Bocca frères, Ch. Clausen. p. 27.
- Suppes 1960, Hatcher 2014 Suppes, Patrick (1960). Axiomatic Set Theory. Dover Publications. ISBN 0-486-61630-4. Derives the Peano axioms from ZFC Hatcher, William S. (2014) [1982]. The Logical Foundations of Mathematics. Elsevier. ISBN 978-1-4831-8963-5. Derives the Peano axioms (called S) from several axiomatic set theories and from category theory.
- Tarski & Givant 1987, Section 7.6. Tarski, Alfred; Givant, Steven (1987). A Formalization of Set Theory without Variables. AMS Colloquium Publications. Vol. 41. American Mathematical Society. ISBN 978-0-8218-1041-5.
- 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. New York, Humanities Press. - 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. p. 133. ISBN 978-0-691-15271-4.
books.google.com
- Grassmann 1861. Grassmann, Hermann Günther (1861). Lehrbuch der Arithmetik für höhere Lehranstalten. Verlag von Theod. Chr. Fr. Enslin.
- Shields 1997. Shields, Paul (1997). "3. Peirce's Axiomatization of Arithmetic". In Houser, Nathan; Roberts, Don D.; Van Evra, James (eds.). Studies in the Logic of Charles Sanders Peirce. Indiana University Press. pp. 43–52. ISBN 0-253-33020-3.
- Suppes 1960, Hatcher 2014 Suppes, Patrick (1960). Axiomatic Set Theory. Dover Publications. ISBN 0-486-61630-4. Derives the Peano axioms from ZFC Hatcher, William S. (2014) [1982]. The Logical Foundations of Mathematics. Elsevier. ISBN 978-1-4831-8963-5. Derives the Peano axioms (called S) from several axiomatic set theories and from category theory.
- 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. New York, Humanities Press. - 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. p. 133. ISBN 978-0-691-15271-4. - Partee, Ter Meulen & Wall 2012, p. 215. Partee, Barbara; Ter Meulen, Alice; Wall, Robert (2012). Mathematical Methods in Linguistics. Springer. ISBN 978-94-009-2213-6.
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- Wang 1957, pp. 145, 147, "It is rather well-known, through Peano's own acknowledgement, that Peano […] made extensive use of Grassmann's work in his development of the axioms. It is not so well-known that Grassmann had essentially the characterization of the set of all integers, now customary in texts of modern algebra, that it forms an ordered integral domain in wihich each set of positive elements has a least member. […] [Grassmann's book] was probably the first serious and rather successful attempt to put numbers on a more or less axiomatic basis.". Wang, Hao (June 1957). "The Axiomatization of Arithmetic". The Journal of Symbolic Logic. 22 (2). Association for Symbolic Logic: 145–158. doi:10.2307/2964176. JSTOR 2964176. S2CID 26896458.
- Peirce 1881. Peirce, C. S. (1881). "On the Logic of Number". American Journal of Mathematics. 4 (1): 85–95. doi:10.2307/2369151. JSTOR 2369151. MR 1507856.
- Hilbert 1902. Hilbert, David (1902). "Mathematische Probleme" [Mathematical Problems]. Bulletin of the American Mathematical Society. 8 (10). Translated by Winton, Maby: 437–479. doi:10.1090/s0002-9904-1902-00923-3.
- Gödel 1931. Gödel, Kurt (1931). "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I" (PDF). Monatshefte für Mathematik. 38. See On Formally Undecidable Propositions of Principia Mathematica and Related Systems for details on English translations.: 173–198. doi:10.1007/bf01700692. S2CID 197663120. Archived from the original (PDF) on 2018-04-11. Retrieved 2013-10-31.
- Gödel 1958 Gödel, Kurt (1958). "Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes". Dialectica. 12 (3–4). Reprinted in English translation in 1990. Gödel's Collected Works, Vol II. Solomon Feferman et al., eds. Oxford University Press: 280–287. doi:10.1111/j.1746-8361.1958.tb01464.x.
- Gentzen 1936 Gentzen, Gerhard (1936). "Die Widerspruchsfreiheit der reinen Zahlentheorie". Mathematische Annalen. 112. Reprinted in English translation in his 1969 Collected works, M. E. Szabo, ed.: 132–213. doi:10.1007/bf01565428. S2CID 122719892.
- 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. JSTOR 2695030. MR 1833464. S2CID 2822314.
- Harsanyi (1983). Harsanyi, John C. (1983). "Mathematics, the Empirical Facts, and Logical Necessity". In Hempel, Carl G.; Putnam, Hilary; Essler, Wilhelm K. (eds.). Methodology, Epistemology, and Philosophy of Science. pp. 167–192. doi:10.1007/978-94-015-7676-5_8. ISBN 978-90-481-8389-0. S2CID 121297669.
harvard.edu
math.harvard.edu
- Gerardo con Diaz, Mathematical Induction Archived 2 May 2013 at the Wayback Machine, Harvard University
illinois.edu
courses.engr.illinois.edu
- Meseguer & Goguen 1986, sections 2.3 (p. 464) and 4.1 (p. 471). Meseguer, José; Goguen, Joseph A. (Dec 1986). "Initiality, induction, and computability". In Maurice Nivat and John C. Reynolds (ed.). Algebraic Methods in Semantics (PDF). Cambridge: Cambridge University Press. pp. 459–541. ISBN 978-0-521-26793-9.
jstor.org
- Wang 1957, pp. 145, 147, "It is rather well-known, through Peano's own acknowledgement, that Peano […] made extensive use of Grassmann's work in his development of the axioms. It is not so well-known that Grassmann had essentially the characterization of the set of all integers, now customary in texts of modern algebra, that it forms an ordered integral domain in wihich each set of positive elements has a least member. […] [Grassmann's book] was probably the first serious and rather successful attempt to put numbers on a more or less axiomatic basis.". Wang, Hao (June 1957). "The Axiomatization of Arithmetic". The Journal of Symbolic Logic. 22 (2). Association for Symbolic Logic: 145–158. doi:10.2307/2964176. JSTOR 2964176. S2CID 26896458.
- Peirce 1881. Peirce, C. S. (1881). "On the Logic of Number". American Journal of Mathematics. 4 (1): 85–95. doi:10.2307/2369151. JSTOR 2369151. MR 1507856.
- 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. JSTOR 2695030. MR 1833464. S2CID 2822314.
semanticscholar.org
api.semanticscholar.org
- Wang 1957, pp. 145, 147, "It is rather well-known, through Peano's own acknowledgement, that Peano […] made extensive use of Grassmann's work in his development of the axioms. It is not so well-known that Grassmann had essentially the characterization of the set of all integers, now customary in texts of modern algebra, that it forms an ordered integral domain in wihich each set of positive elements has a least member. […] [Grassmann's book] was probably the first serious and rather successful attempt to put numbers on a more or less axiomatic basis.". Wang, Hao (June 1957). "The Axiomatization of Arithmetic". The Journal of Symbolic Logic. 22 (2). Association for Symbolic Logic: 145–158. doi:10.2307/2964176. JSTOR 2964176. S2CID 26896458.
- Gödel 1931. Gödel, Kurt (1931). "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I" (PDF). Monatshefte für Mathematik. 38. See On Formally Undecidable Propositions of Principia Mathematica and Related Systems for details on English translations.: 173–198. doi:10.1007/bf01700692. S2CID 197663120. Archived from the original (PDF) on 2018-04-11. Retrieved 2013-10-31.
- Gentzen 1936 Gentzen, Gerhard (1936). "Die Widerspruchsfreiheit der reinen Zahlentheorie". Mathematische Annalen. 112. Reprinted in English translation in his 1969 Collected works, M. E. Szabo, ed.: 132–213. doi:10.1007/bf01565428. S2CID 122719892.
- 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. JSTOR 2695030. MR 1833464. S2CID 2822314.
- Harsanyi (1983). Harsanyi, John C. (1983). "Mathematics, the Empirical Facts, and Logical Necessity". In Hempel, Carl G.; Putnam, Hilary; Essler, Wilhelm K. (eds.). Methodology, Epistemology, and Philosophy of Science. pp. 167–192. doi:10.1007/978-94-015-7676-5_8. ISBN 978-90-481-8389-0. S2CID 121297669.
sfu.ca
- Matt DeVos, Mathematical Induction, Simon Fraser University
w-k-essler.de
- Gödel 1931. Gödel, Kurt (1931). "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I" (PDF). Monatshefte für Mathematik. 38. See On Formally Undecidable Propositions of Principia Mathematica and Related Systems for details on English translations.: 173–198. doi:10.1007/bf01700692. S2CID 197663120. Archived from the original (PDF) on 2018-04-11. Retrieved 2013-10-31.
web.archive.org
- Gerardo con Diaz, Mathematical Induction Archived 2 May 2013 at the Wayback Machine, Harvard University
- Gödel 1931. Gödel, Kurt (1931). "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I" (PDF). Monatshefte für Mathematik. 38. See On Formally Undecidable Propositions of Principia Mathematica and Related Systems for details on English translations.: 173–198. doi:10.1007/bf01700692. S2CID 197663120. Archived from the original (PDF) on 2018-04-11. Retrieved 2013-10-31.
worldcat.org
- Hermes 1973, VI.4.3, presenting a theorem of Thoralf Skolem Hermes, Hans (1973). Introduction to Mathematical Logic. Hochschultext. Springer. ISBN 3-540-05819-2. ISSN 1431-4657.
- Hermes 1973, VI.3.1. Hermes, Hans (1973). Introduction to Mathematical Logic. Hochschultext. Springer. ISBN 3-540-05819-2. ISSN 1431-4657.