Natural Numbers and Peano’s Axioms<\/title> <\/head> <body> <\/p>\n<div style=\"padding:20px;\">\n<center><\/p>\n<h1>Natural Numbers and Peano’s Axioms<\/h1>\nSUMMARY<\/b> \nThis class covers natural numbers and how they are defined using Peano’s axioms: a series of mathematical principles that establish their fundamental properties. It also explains how symbols are used to represent the successors of natural numbers, their symbolic representation, and the use of the principle of mathematical induction for conducting inductive proofs.<\/em><\/p>\nLEARNING OBJECTIVES<\/b><\/p>\n<\/center><\/p>\n<ol>\n<li>Understand<\/strong> Peano’s axioms for the formulation of natural numbers.<\/li>\n<li>Understand<\/strong> the formulation of the symbolic representation of natural numbers.<\/li>\n<\/ol>\n<center><\/p>\nINDEX<\/strong><\/p>\n<a href=\"#1\">The Peano’s axioms<\/strong><\/a> \n<a href=\"#2\">The principle of induction in natural numbers<\/strong><\/a> \n<a href=\"#3\">Commentary on demonstrations<\/a><\/p>\n<\/center><\/p>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/w-BznjX88No\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/div>\n<a name=\"1\"><\/a><\/p>\n<h2>The Peano’s Axioms<\/h2>\nThe Natural Numbers<\/em>, also known as positive integers<\/em>, are those we use for counting and measuring. They naturally arise in the operation of counting, which is the simplest arithmetic operation. These numbers are defined through the Peano’s axioms<\/em><\/strong>, a set of mathematical principles that establish how these numbers function.<\/p>\n<ol style=\"text-align: justify;\">\n<li>\u00ab1<\/span>\u00bb is a natural number.<\/li>\n<li>If n<\/span> is a natural number, then its successor S(n)<\/span> is also a natural number.<\/li>\n<li>\u00ab1<\/span>\u00bb is not the successor of any natural number.<\/li>\n<li>If S(n) = S(m)<\/span>, then n=m<\/span>.<\/li>\n<li>If 1<\/span> belongs to some set A<\/span>; and if, given any k<\/span> in A<\/span>, S(k)<\/span> is also in A<\/span>, then A<\/span> is the set of natural numbers denoted by \\mathbb{N}<\/span>.<\/li>\n<\/ol>\nWhen studying Peano’s axioms, we realize that the symbol \u00ab1<\/span>\u00bb is just a representation used to indicate a specific natural number. This number is the one that meets these properties. Just as 1<\/span> represents the \u00abfirst natural\u00bb, we also use symbols (which are familiar to us) to represent its successors. <\/p>\n<ul style=\"text-align: justify;\">\n<li>2=S(1)<\/span><\/li>\n<li>3=S(2)<\/span><\/li>\n<li>4=S(3) \\\\ \\vdots<\/span><\/li>\n<\/ul>\nand so on. In this way, the symbols 1<\/span>, 2<\/span>, 3<\/span>, etc… are abstract entities representing the different successors of 1<\/span>. The collection of all these objects are the natural numbers, which we represent as:<\/p>\n\\mathbb{N}=\\{1,2,3,4,\\cdots \\}<\/span>\nIt is also said that natural numbers are arranged in a sequence, the sequence of natural numbers:<\/p>\n1,2,3,4,5,6,7,8,9,10,11,12, \\cdots <\/span>\n<a name=\"2\"><\/a><\/p>\n<h2>The Principle of Induction in Natural Numbers<\/h2>\nAn important aspect of natural numbers is that there is always a number after each one, which means there are infinite. We can intuit this from the fifth axiom, or principle of induction,<\/strong> which is expressed as follows:<\/p>\nIf a property holds for 1<\/span>; and if given that it holds for any natural number k<\/span>, it also holds for the next S(k)<\/span>; then such property holds for all natural numbers.<\/em><\/p>\nThe principle of induction provides, in addition to a foundational basis, a useful tool to prove if a property holds for natural numbers. To understand this, let’s see a simple example:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify;\">\nEXAMPLE:<\/strong><\/span> Through the principle of induction, it can be demonstrated that every natural number is different from its successor.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: justify;\">\nWhile this is obvious, it helps to understand the procedure when proving through induction.<\/p>\nProof:<\/strong><\/p>\n<ul>\n<li>\nIt’s clear that 1<\/span> is different from S(1)=2<\/span>. This is the initial step,<\/strong> where we verify that the property holds for the first element.<\/p>\n<\/li>\n<li>\nAssume the property holds for any k<\/span>, that is, k\\neq S(k)<\/span>; what we will do is prove that from this it is also verified for S(k)<\/span> (i.e., it is also verified that S(k)\\neq S(S(k))<\/span>. This is the inductive step.<\/strong> If these two steps are completed, then it is said that the induction is complete and the property holds for all natural numbers.<\/p>\n<\/p>\n[1]<\/strong> To start, note that S(k) \\neq k,<\/span> is equivalent to saying \\neg [k=S(k)]<\/span>.<\/p>\n[2]<\/strong> But since both k<\/span> and S(k)<\/span> are natural numbers, by axiom 2 we can say they both have successors: S(k)<\/span> and S(S(k))<\/span>, respectively. Both are also natural numbers.<\/p>\n[3]<\/strong> Then, by axiom 4 we can say that: S(k) = S(S(k))<\/span> implies that k = S(k)<\/span>. We can express this as:<\/p>\n\\left[ S(k) = S(S(k)) \\right] \\rightarrow \\left[k = S(k)\\right]<\/span>\nwhich by contrapositive of the implication, is equivalent to saying:<\/p>\n\\neg \\left[k = S(k)\\right] \\rightarrow \\neg \\left[ S(k) = S(S(k)) \\right] <\/span>\n[4]<\/strong> Finally, applying a modus ponens between this last expression and the one obtained in step [1]<\/strong>, we get<\/p>\n\\neg \\left[ S(k) = S(S(k)) \\right] <\/span>\nwhich is the same as saying<\/p>\n S(k) \\neq S(S(k)) <\/span>\nAnd therefore, we have shown that if it is verified that S(k) \\neq k,<\/span>, then it is also verified that S(k) \\neq S(S(k))<\/span>; as it is also evident that 1\\neq 2<\/span>, the induction is complete and it can be written that<\/p>\n\\left(\\forall n\\in\\mathbb{N}\\right)\\left(n \\neq S(n)\\right) <\/span>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<a name=\"3\"><\/a><\/p>\n<h3>Comment on Proofs<\/h3>\nWhile the property stated in the example is quite obvious, it’s common in mathematics for proofs not to maintain this obviousness. This proof we just saw is an example of what is typically done when working mathematically. To support your understanding of the deduction techniques inherent in mathematics, I recommend you review the materials designed for the <a href=\"http:\/\/toposuranos.com\/material\/en\/category\/mathematics\/mathematical-logic\/propositional-logic\/\" target=\"_blank\" rel=\"noopener\">mathematical logic<\/strong><\/a> course.<\/p>\n<\/body> \n<\/html><\/p>\n","protected":false},"excerpt":{"rendered":"Natural Numbers and Peano’s Axioms Natural Numbers and Peano’s Axioms SUMMARY This class covers natural numbers and how they are defined using Peano’s axioms: a series of mathematical principles that establish their fundamental properties. It also explains how symbols are used to represent the successors of natural numbers, their symbolic representation, and the use of […]<\/p>\n","protected":false},"author":1,"featured_media":25048,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":23,"footnotes":""},"categories":[583,1031,567],"tags":[],"class_list":["post-24980","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-and-geometry","category-general-algebra","category-mathematics"],"yoast_head":"\n<title>Natural Numbers and Peano's Axioms - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Discover the natural numbers through Peano's axioms, how they are symbolically represented, and the principle of mathematical induction.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/en\/natural-numbers-and-peanos-axioms\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Natural Numbers and Peano's Axioms\" \/>\n<meta property=\"og:description\" content=\"Discover the natural numbers through Peano's axioms, how they are symbolically represented, and the principle of mathematical induction.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/en\/natural-numbers-and-peanos-axioms\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-02-28T00:00:50+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-03-02T19:49:54+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/quesonlosnaturales-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Natural Numbers and Peano's Axioms\" \/>\n<meta name=\"twitter:description\" content=\"Discover the natural numbers through Peano's axioms, how they are symbolically represented, and the principle of mathematical induction.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/quesonlosnaturales.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/natural-numbers-and-peanos-axioms\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/natural-numbers-and-peanos-axioms\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Natural Numbers and Peano’s Axioms\",\"datePublished\":\"2021-02-28T00:00:50+00:00\",\"dateModified\":\"2025-03-02T19:49:54+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/natural-numbers-and-peanos-axioms\\\/\"},\"wordCount\":938,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/natural-numbers-and-peanos-axioms\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2021\\\/03\\\/numerosnaturales-1.jpg\",\"articleSection\":[\"Algebra and Geometry\",\"General Algebra\",\"Mathematics\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/natural-numbers-and-peanos-axioms\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/natural-numbers-and-peanos-axioms\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/natural-numbers-and-peanos-axioms\\\/\",\"name\":\"Natural Numbers and Peano's Axioms - 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{"id":24980,"date":"2021-02-28T00:00:50","date_gmt":"2021-02-28T00:00:50","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=24980"},"modified":"2025-03-02T19:49:54","modified_gmt":"2025-03-02T19:49:54","slug":"natural-numbers-and-peanos-axioms","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/natural-numbers-and-peanos-axioms\/","title":{"rendered":"Natural Numbers and Peano’s Axioms"},"content":{"rendered":"

\n Natural Numbers and Peano’s Axioms<\/title> <\/head> <body> <\/p>\n<div style=\"padding:20px;\">\n<center><\/p>\n<h1>Natural Numbers and Peano’s Axioms<\/h1>\nSUMMARY<\/b> \nThis class covers natural numbers and how they are defined using Peano’s axioms: a series of mathematical principles that establish their fundamental properties. It also explains how symbols are used to represent the successors of natural numbers, their symbolic representation, and the use of the principle of mathematical induction for conducting inductive proofs.<\/em><\/p>\nLEARNING OBJECTIVES<\/b><\/p>\n<\/center><\/p>\n<ol>\n<li>Understand<\/strong> Peano’s axioms for the formulation of natural numbers.<\/li>\n<li>Understand<\/strong> the formulation of the symbolic representation of natural numbers.<\/li>\n<\/ol>\n<center><\/p>\nINDEX<\/strong><\/p>\n<a href=\"#1\">The Peano’s axioms<\/strong><\/a> \n<a href=\"#2\">The principle of induction in natural numbers<\/strong><\/a> \n<a href=\"#3\">Commentary on demonstrations<\/a><\/p>\n<\/center><\/p>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/w-BznjX88No\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/div>\n<a name=\"1\"><\/a><\/p>\n<h2>The Peano’s Axioms<\/h2>\nThe Natural Numbers<\/em>, also known as positive integers<\/em>, are those we use for counting and measuring. They naturally arise in the operation of counting, which is the simplest arithmetic operation. These numbers are defined through the Peano’s axioms<\/em><\/strong>, a set of mathematical principles that establish how these numbers function.<\/p>\n<ol style=\"text-align: justify;\">\n<li>\u00ab1<\/span>\u00bb is a natural number.<\/li>\n<li>If n<\/span> is a natural number, then its successor S(n)<\/span> is also a natural number.<\/li>\n<li>\u00ab1<\/span>\u00bb is not the successor of any natural number.<\/li>\n<li>If S(n) = S(m)<\/span>, then n=m<\/span>.<\/li>\n<li>If 1<\/span> belongs to some set A<\/span>; and if, given any k<\/span> in A<\/span>, S(k)<\/span> is also in A<\/span>, then A<\/span> is the set of natural numbers denoted by \\mathbb{N}<\/span>.<\/li>\n<\/ol>\nWhen studying Peano’s axioms, we realize that the symbol \u00ab1<\/span>\u00bb is just a representation used to indicate a specific natural number. This number is the one that meets these properties. Just as 1<\/span> represents the \u00abfirst natural\u00bb, we also use symbols (which are familiar to us) to represent its successors. <\/p>\n<ul style=\"text-align: justify;\">\n<li>2=S(1)<\/span><\/li>\n<li>3=S(2)<\/span><\/li>\n<li>4=S(3) \\\\ \\vdots<\/span><\/li>\n<\/ul>\nand so on. In this way, the symbols 1<\/span>, 2<\/span>, 3<\/span>, etc… are abstract entities representing the different successors of 1<\/span>. The collection of all these objects are the natural numbers, which we represent as:<\/p>\n\\mathbb{N}=\\{1,2,3,4,\\cdots \\}<\/span>\nIt is also said that natural numbers are arranged in a sequence, the sequence of natural numbers:<\/p>\n1,2,3,4,5,6,7,8,9,10,11,12, \\cdots <\/span>\n<a name=\"2\"><\/a><\/p>\n<h2>The Principle of Induction in Natural Numbers<\/h2>\nAn important aspect of natural numbers is that there is always a number after each one, which means there are infinite. We can intuit this from the fifth axiom, or principle of induction,<\/strong> which is expressed as follows:<\/p>\nIf a property holds for 1<\/span>; and if given that it holds for any natural number k<\/span>, it also holds for the next S(k)<\/span>; then such property holds for all natural numbers.<\/em><\/p>\nThe principle of induction provides, in addition to a foundational basis, a useful tool to prove if a property holds for natural numbers. To understand this, let’s see a simple example:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify;\">\nEXAMPLE:<\/strong><\/span> Through the principle of induction, it can be demonstrated that every natural number is different from its successor.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: justify;\">\nWhile this is obvious, it helps to understand the procedure when proving through induction.<\/p>\nProof:<\/strong><\/p>\n<ul>\n<li>\nIt’s clear that 1<\/span> is different from S(1)=2<\/span>. This is the initial step,<\/strong> where we verify that the property holds for the first element.<\/p>\n<\/li>\n<li>\nAssume the property holds for any k<\/span>, that is, k\\neq S(k)<\/span>; what we will do is prove that from this it is also verified for S(k)<\/span> (i.e., it is also verified that S(k)\\neq S(S(k))<\/span>. This is the inductive step.<\/strong> If these two steps are completed, then it is said that the induction is complete and the property holds for all natural numbers.<\/p>\n<\/p>\n[1]<\/strong> To start, note that S(k) \\neq k,<\/span> is equivalent to saying \\neg [k=S(k)]<\/span>.<\/p>\n[2]<\/strong> But since both k<\/span> and S(k)<\/span> are natural numbers, by axiom 2 we can say they both have successors: S(k)<\/span> and S(S(k))<\/span>, respectively. Both are also natural numbers.<\/p>\n[3]<\/strong> Then, by axiom 4 we can say that: S(k) = S(S(k))<\/span> implies that k = S(k)<\/span>. We can express this as:<\/p>\n\\left[ S(k) = S(S(k)) \\right] \\rightarrow \\left[k = S(k)\\right]<\/span>\nwhich by contrapositive of the implication, is equivalent to saying:<\/p>\n\\neg \\left[k = S(k)\\right] \\rightarrow \\neg \\left[ S(k) = S(S(k)) \\right] <\/span>\n[4]<\/strong> Finally, applying a modus ponens between this last expression and the one obtained in step [1]<\/strong>, we get<\/p>\n\\neg \\left[ S(k) = S(S(k)) \\right] <\/span>\nwhich is the same as saying<\/p>\n S(k) \\neq S(S(k)) <\/span>\nAnd therefore, we have shown that if it is verified that S(k) \\neq k,<\/span>, then it is also verified that S(k) \\neq S(S(k))<\/span>; as it is also evident that 1\\neq 2<\/span>, the induction is complete and it can be written that<\/p>\n\\left(\\forall n\\in\\mathbb{N}\\right)\\left(n \\neq S(n)\\right) <\/span>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<a name=\"3\"><\/a><\/p>\n<h3>Comment on Proofs<\/h3>\nWhile the property stated in the example is quite obvious, it’s common in mathematics for proofs not to maintain this obviousness. This proof we just saw is an example of what is typically done when working mathematically. To support your understanding of the deduction techniques inherent in mathematics, I recommend you review the materials designed for the <a href=\"http:\/\/toposuranos.com\/material\/en\/category\/mathematics\/mathematical-logic\/propositional-logic\/\" target=\"_blank\" rel=\"noopener\">mathematical logic<\/strong><\/a> course.<\/p>\n<\/body> \n<\/html><\/p>\n","protected":false},"excerpt":{"rendered":"Natural Numbers and Peano’s Axioms Natural Numbers and Peano’s Axioms SUMMARY This class covers natural numbers and how they are defined using Peano’s axioms: a series of mathematical principles that establish their fundamental properties. It also explains how symbols are used to represent the successors of natural numbers, their symbolic representation, and the use of […]<\/p>\n","protected":false},"author":1,"featured_media":25048,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":23,"footnotes":""},"categories":[583,1031,567],"tags":[],"class_list":["post-24980","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-and-geometry","category-general-algebra","category-mathematics"],"yoast_head":"\n<title>Natural Numbers and Peano's Axioms - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Discover the natural numbers through Peano's axioms, how they are symbolically represented, and the principle of mathematical induction.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/en\/natural-numbers-and-peanos-axioms\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Natural Numbers and Peano's Axioms\" \/>\n<meta property=\"og:description\" content=\"Discover the natural numbers through Peano's axioms, how they are symbolically represented, and the principle of mathematical induction.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/en\/natural-numbers-and-peanos-axioms\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-02-28T00:00:50+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-03-02T19:49:54+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/quesonlosnaturales-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Natural Numbers and Peano's Axioms\" \/>\n<meta name=\"twitter:description\" content=\"Discover the natural numbers through Peano's axioms, how they are symbolically represented, and the principle of mathematical induction.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/quesonlosnaturales.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/natural-numbers-and-peanos-axioms\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/natural-numbers-and-peanos-axioms\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Natural Numbers and Peano’s Axioms\",\"datePublished\":\"2021-02-28T00:00:50+00:00\",\"dateModified\":\"2025-03-02T19:49:54+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/natural-numbers-and-peanos-axioms\\\/\"},\"wordCount\":938,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/natural-numbers-and-peanos-axioms\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2021\\\/03\\\/numerosnaturales-1.jpg\",\"articleSection\":[\"Algebra and Geometry\",\"General Algebra\",\"Mathematics\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/natural-numbers-and-peanos-axioms\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/natural-numbers-and-peanos-axioms\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/natural-numbers-and-peanos-axioms\\\/\",\"name\":\"Natural Numbers and Peano's Axioms - 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