20.500.12556/RUL-87122 Aksiomatska konstrukcija naravnih števil magistrsko delo Axiomatic construction of natural numbers V magistrskem delu so najprej predstavljena naravna števila, in sicer smo jih vpeljali preko Peanovih aksiomov. Z vsemi petimi aksiomi postopoma prikažemo računski operaciji seštevanje in množenje ter njune osnovne lastnosti (nevtralni element, komutativnost, asociativnost, distributivnost). Predstavljena so tudi rekurzivno definirana zaporedja, urejenost naravnih števil ter odštevanje in deljenje, ki pa sta le delno definirani operaciji v naravnih številih. V nadaljevanju je velik poudarek na natančnih dokazih osnovnih lastnosti naravnih števil le s pomočjo Peanovih aksiomov. V poglavju o množicah smo se v glavnem posvetili sistemu ZFC aksiomov, ki so ime dobili po matematikih Zermelu in Fraenkelu ter aksiomu izbire (C). S pomočjo aksiomatske teorije množic (predvsem aksioma o neskončnosti) naravna števila vpeljemo kot množico, v kateri pokažemo veljavnost Peanovih aksiomov. V zadnjem delu smo vpeljali še cela in racionalna števila kot kvocientni množici kartezičnega produkta N×N oziroma Z×Z\\{0}. The natural numbers are presented first in the master's thesis. We introduced them through Pean axioms. With all five axioms we gradually show the arithmetic operations of addition and multiplication and their basic characteristics (neutral element, commutative, associative and distributive properties). The recursively defined sequences, order of natural numbers and subtraction and division, which are only partially defined operations in natural numbers, are also presented. A great emphasis is on accurate proofs of basic properties of natural numbers only with the help of Pean axioms. In the chapter about sets we focused our attention to the system of ZFC axioms, which got their names after mathematicians Zermel and Fraenkel and the axiom of choice (C). With the help of axiomatic of the theory of sets (mainly axiom of infinity) we introduce natural numbers like a set in which we show validity of Pean axioms. In the last part of the thesis we also introduced integers and rational numbers like quotient sets of the Cartesian product N×N or Z×Z\\{0}. realna števila Peanovi aksiomi rekurzivno zaporedje teorija množic ZFC aksiomi cela števila racionalna števila mathematics matematika true false false [Š. Zobavnik] Slovenski jezik Angleški jezik Magistrsko delo/naloga 2016-11-25 02:20:09 2017-09-04 15:34:15 2025-04-17 04:52:44 0000-00-00 00:00:00 2016 0 0 49 str. 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 511(043.2) 11335753 Aksiomatska_konstrukcija_naravnih_stevil.pdf Aksiomatska_konstrukcija_naravnih_stevil.pdf 1 5F480C751AA79A44DC31C2410C66CA81 57689bec4d1a89dea5777c15f9af6b45372bb1e926855ef0eec6a4a625fc4e0e 8a7b0bfe-1aac-11f0-b232-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=205163 Pedagoška fakulteta 0 0 0