[過去ﾛｸﾞ] 現代数学の系譜 工学物理雑談 古典ガロア理論も読む62 (1002ﾚｽ)

現代数学の系譜 工学物理雑談 古典ガロア理論も読む62 http://rio2016.5ch.net/test/read.cgi/math/1551963737/

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175: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2019/03/11(月) 07:59:04.77 ID:NUGiaq8/ >>172 （参考追加） https://en.wikipedia.org/wiki/Peano_axioms Peano axioms (抜粋) 4 Models 4.1 Set-theoretic models 4.2 Interpretation in category theory 5 Nonstandard models https://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic Non-standard model of arithmetic (抜粋) In mathematical logic, a non-standard model of arithmetic is a model of (first-order) Peano arithmetic that contains non-standard numbers. The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment. The construction of such models is due to Thoralf Skolem (1934). つづく http://rio2016.5ch.net/test/read.cgi/math/1551963737/175

176: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2019/03/11(月) 07:59:44.48 ID:NUGiaq8/ >>175 つづき Contents 1 Existence 1.1 From the compactness theorem 1.2 From the incompleteness theorems 1.2.1 Arithmetic unsoundness for models with ~G true 1.3 From an ultraproduct 2 Structure of countable non-standard models References ・Boolos, G., and Jeffrey, R. 1974. Computability and Logic, Cambridge University Press. ISBN 0-521-38923-2 ・Thoralf Skolem (1934). "Uber die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzahlbar unendlich vieler Aussagen mit ausschlieslich Zahlenvariablen" (PDF). Fundamenta Mathematicae (in German). 23 (1): 150?161. http://matwbn.icm.edu.pl/ksiazki/fm/fm23/fm23115.pdf Citations 1 Andrey Bovykin and Richard Kaye Order-types of models of Peano arithmetic: a short survey June 14, 2001 http://web.mat.bham.ac.uk/R.W.Kaye/papers/survey6.pdf 2 Andrey Bovykin On order-types of models of arithmetic thesis submitted to the University of Birmingham for the degree of Ph.D. in the Faculty of Science 13 April 2000 http://logic.pdmi.ras.ru/~andrey/phd.pdf 3 Fred Landman LINEAR ORDERS, DISCRETE, DENSE, AND CONTINUOUS ? includes proof that Q is the only countable dense linear order. http://www.tau.ac.il/~landman/Online_Class_Notes_file/Boolean/1%20%20Linear%20orders,%20discrete,%20dense%20and%20continuous.pdf (引用終り) 以上 http://rio2016.5ch.net/test/read.cgi/math/1551963737/176

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