The purpose of this paper is to develop the Peano postulates from a weaker axiom system than the system used by John L. Kelley in General Topology. The axiom of regularity which states "If X is a non-empty set, then there is a member Y of X such that the intersection of X and Y is empty." is not assumed in this thesis. The axiom of amalgamation which states "If X is a set, then the union of the elements of X is a set." is also not assumed. All other axioms used by Kelley relevant to the Peano postulates are assumed. The word class is never used in the thesis, though the variables can be interpreted as classes.

[pt] Sabemos pelo Teorema da Incompletude de Godel que existem afirmações verdadeiras sobre números naturais que não podem ser demonstradas na aritmética de Peano. Paris e Harrington deram um exemplo de uma variação do Teorema de Ramsey finito que não pode ser demonstrada em aritmética de Peano apesar de ser facilmente demonstrável na Teoria de Conjuntos usual. Este é geralmente considerado o primeiro exemplo matematicamente natural de uma sentença indecidível. Além da demonstração original, apresentamos nessa dissertação outra usando Teoria de Modelos. / [en] From Godel’s Incompleteness Theorem we know that there are true sentences about natural numbers which can not be proved in Peano Arithmetic. Paris and Harrington gave an example of a variation of the finite Ramsey Theorem which can not be proved in Peano Arithmetic although it can be easily proved in usual Set Theory. This is usually considered the first example of a mathematically natural undecidable sentence. Besides the original proof, another one, using Model Theory, is presented in this dissertation.

Thesis (M.S.)--Brigham Young University. Dept. of Mathematics, 2005. / Includes bibliographical references (leaf 15).

SOUSA, Pedro Sérgio Sales de. A construção dos números naturais: um foco nas quatro operações fundamentais. 2014. 40 f. Dissertação (Mestrado em Matemática em Rede Nacional) - Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2014. / Submitted by Erivan Almeida (eneiro@bol.com.br) on 2015-01-12T16:34:08Z No. of bitstreams: 1 2014_dis_psssousa.pdf: 1314798 bytes, checksum: d58f97e2efdcfc7a1a8d07e1edb49b0b (MD5) / Approved for entry into archive by Rocilda Sales(rocilda@ufc.br) on 2015-01-15T12:57:36Z (GMT) No. of bitstreams: 1 2014_dis_psssousa.pdf: 1314798 bytes, checksum: d58f97e2efdcfc7a1a8d07e1edb49b0b (MD5) / Made available in DSpace on 2015-01-15T12:57:36Z (GMT). No. of bitstreams: 1 2014_dis_psssousa.pdf: 1314798 bytes, checksum: d58f97e2efdcfc7a1a8d07e1edb49b0b (MD5) Previous issue date: 2014 / This paper aims to present the construction of the natural numbers and the axiomatic definition with respect to the four fundamental operations for students and teachers of elementary school.To this was presented a sequence initially addressing on the study of mathematics, the concept of mathematics, mathematical knowledge and a mathematical brief history to see how mathematical theories and practices are designed, developed and used in a specific context of each era. The second moment was described the construction of natural numbers through the Peano axioms, continuing with the rigorous definition of each operation and ending with the order relation in the set of natural numbers. / O presente trabalho tem como objetivo apresentar a construção dos números naturais e a definição axiomática no que diz respeito às quatro operações fundamentais para alunos e professores do ensino fundamental. Para isso foi apresentado uma sequência abordando inicialmente as considerações sobre o estudo da Matemática, o conceito de Matemática, o saber matemático e um breve histórico matemático para se perceber como teorias e práticas matemáticas foram criadas, desenvolvidas e utilizadas num contexto específico de cada época. No segundo momento foi descrita a construção dos números naturais através dos axiomas de Peano, prosseguindo com a definição rigorosa de cada operação e finalizando com a relação de ordem no conjunto dos números naturais.

This paper contains detailed description of two consistency proofs, which state that in the system called Peano arithmetic no contradiction can be obtained. The proofs were first published in 1936 and 1938 by the German mathematician Gerhard Gentzen. For the purpose of this paper, the proofs were read and studied from the original articles called "Die Widerspruchsfreiheit der reinen Zahlentheorie" and "Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie". The first mentioned proof is interesting from the historical point of view. Gentzen used a natural deduction sequent calculus and ordinal numbers in an unusual form he invented. The second proof is similar to the consistency proof, which is commonly known as a consistency proof for Peano arithmetic nowadays.

The purpose of this thesis is to present an axiomatic foundation for the development of the natural numbers from two points of view. It makes no claim at originality other than at the point of organization and presentation of previously developed works.

Let X be a planar or one-dimensional Peano continuum. Let E be a Hawaiian Earring with fundamental group H. We show that every homomorphism from H to the fundamental group of X is conjugate to a homomorphism which is induced by a continuous function.

<p> This paper surveys. the fundamental theory of universal topological and uniform algebras and relates the former to topological partial algebras. In particular, the theory of topological and uniform Peano algebras is described and some other universal mapping questions analysed and solved. In addition a theory of compact extensions is developed for topological algebras.</p> / Thesis / Doctor of Philosophy (PhD)

This thesis covers the development of the natural numbers by means of the peano postulates.

O presente trabalho tem como objetivo apresentar a construÃÃo dos nÃmeros naturais e a definiÃÃo axiomÃtica no que diz respeito Ãs quatro operaÃÃes fundamentais para alunos e professores do ensino fundamental. Para isso foi apresentado uma sequÃncia abordando inicialmente as consideraÃÃes sobre o estudo da MatemÃtica, o conceito de MatemÃtica, o saber matemÃtico e um breve histÃrico matemÃtico para se perceber como teorias e prÃticas matemÃticas foram criadas, desenvolvidas e utilizadas num contexto especÃfico de cada Ãpoca. No segundo momento foi descrita a construÃÃo dos nÃmeros naturais atravÃs dos axiomas de Peano, prosseguindo com a definiÃÃo rigorosa de cada operaÃÃo e finalizando com a relaÃÃo de ordem no conjunto dos nÃmeros naturais. / This paper aims to present the construction of the natural numbers and the axiomatic definition with respect to the four fundamental operations for students and teachers of elementary school.To this was presented a sequence initially addressing on the study of mathematics, the concept of mathematics, mathematical knowledge and a mathematical brief history to see how mathematical theories and practices are designed, developed and used in a specific context of each era. The second moment was described the construction of natural numbers through the Peano axioms, continuing with the rigorous definition of each operation and ending with the order relation in the set of natural numbers.