Aksiomatska konstrukcija naravnih števil

magistrsko delo

Špela Zobavnik (Author), Marko Slapar (Mentor)

Abstract

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}.

Keywords

realna števila;Peanovi aksiomi;rekurzivno zaporedje;teorija množic;ZFC aksiomi;cela števila;racionalna števila;

Data

Language:	Slovenian
Year of publishing:	2016
Typology:	2.09 - Master's Thesis
Organization:	UL PEF - Faculty of Education
Publisher:	[Š. Zobavnik]
UDC:	511(043.2)
COBISS:	11335753
Views:	725
Downloads:	169
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Axiomatic construction of natural numbers
Secondary abstract:	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}.
Secondary keywords:	mathematics;matematika;
File type:	application/pdf
Type (COBISS):	Master's thesis/paper
Thesis comment:	Univ. v Ljubljani, Pedagoška fak., Poučevanje, Predmetno poučevanje, Matematika in tehnika
Pages:	49 str.
ID:	9228536