Numeri Naturales et Axiomata Peano<\/title> <\/head> <body> <\/p>\n<div style=\"padding:20px;\"><center><\/p>\n<h1>Numeri Naturales et Axiomata Peano<\/h1>\nCOMPENDIUM<\/b> \nHaec lectio versatur de numeris naturalibus et quomodo per axiomata Peano definiantur: seriem principiorum mathematicorum quae proprietates fundamentales eorum constituunt. Etiam explicat quomodo symbola adhibita sint ad repraesentandos successores numerorum naturalium, quomodo symbolice repraesententur, et usum principii inductionis mathematicae ad probationes inductivas perficiendas.<\/em><\/p>\nPROPOSITA DISCENDI<\/b><\/p>\n<\/center><\/p>\n<ol>\n<li>Intellegere<\/strong> axiomata Peano ad formulationem numerorum naturalium.<\/li>\n<li>Intellegere<\/strong> formulationem repraesentationis symbolicae numerorum naturalium. <\/li>\n<\/ol>\n<center><\/p>\nINDEX<\/strong><\/p>\n<a href=\"#1\">Axiomata Peano pro numeris naturalibus<\/strong><\/a> \n<a href=\"#2\">Principium inductionis in numeris naturalibus<\/strong><\/a> \n<a href=\"#3\">Commentarium de demonstrationibus<\/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>Axiomata Peano pro Numeris Naturalibus<\/h2>\nNumeri Naturales<\/em>, etiam noti ut numeri integri positivii<\/em>, sunt illi quos ad numerandum et metiendum adhibemus. Apparent naturalissime in operatione numerandi, quae est simplicissima ex arithmetica. Hi numeri definiuntur per axiomata Peano<\/em><\/strong>, seriem principiorum mathematicorum quae modum operandum horum numerorum constituunt.<\/p>\n<ol style=\"text-align: justify;\">\n<li>\u00ab1<\/span>\u00bb est numerus naturalis<\/li>\n<li>Si n<\/span> est naturalis, tum eius successor S(n)<\/span> quoque naturalis est.<\/li>\n<li>\u00ab1<\/span>\u00bb non est successor ullius numeri naturalis.<\/li>\n<li>Si S(n) = S(m)<\/span>, tum n=m<\/span>.<\/li>\n<li>Si 1<\/span> ad aliquod collectum A<\/span> pertinet; et si dato k<\/span> quolibet in A<\/span>, etiam S(k)<\/span> in A<\/span> est, tum A<\/span> est collectio numerorum naturalium et notatur \\mathbb{N}<\/span>.<\/li>\n<\/ol>\nDum axiomata Peano examinamus, intellegimus symbolum \u00ab1<\/span>\u00bb revera esse solam repraesentationem adhibitam ad significandum numerum naturalem certum. Hic numerus est is qui has proprietates implet. Sicut 1<\/span> primum naturalem repraesentat, ita quoque symbolis (quae nobis sunt familiares) utimur ad successores eius repraesentandos.<\/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>\net sic porro. Hoc modo, symbola 1<\/span>, 2<\/span>, 3<\/span>, etc., sunt entia abstracta quae diversos successores 1<\/span> repraesentant. Collectio horum omnium obiectorum sunt numeri naturales, quos sic repraesentamus:<\/p>\n\\mathbb{N}=\\{1,2,3,4,\\cdots \\}<\/span>\nEtiam dicitur numeri naturales in successione ordinari, successione numerorum naturalium:<\/p>\n1,2,3,4,5,6,7,8,9,10,11,12, \\cdots <\/span>\n<a name=\"2\"><\/a><\/p>\n<h2>Principium Inductionis in Numeris Naturalibus<\/h2>\nMomentum magnum numerorum naturalium est quod semper est numerus post quemlibet, quod significat infinitos esse numeros naturales. Hoc ex quinto axioma, sive principio inductionis,<\/strong> colligi potest, quod sic exprimitur:<\/p>\nSi aliqua proprietas verificatur pro 1<\/span>; et si, posito quod verificetur pro alio naturali quolibet k<\/span>, etiam verificatur pro sequente S(k)<\/span>; tum talis proprietas verificatur pro omnibus naturalibus.<\/em><\/p>\nPrincipium inductionis praebet, praeter fundamentum numeris naturalibus, instrumentum utile ad demonstrandum si proprietas de numeris naturalibus valeat. Ad hoc inspiciendum, exemplum simplex consideremus:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify;\">\nEXEMPLUM:<\/strong> Per principium inductionis demonstrari potest omnem naturalem diversum esse a suo successore. <\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: justify;\">\nQuamquam hoc videtur manifestum, tamen adiuvat ad intellegendam rationem procedendi cum per inductionem demonstratur. <\/p>\nDemonstratio:<\/strong><\/p>\n<ul>\n<li>\nManifestum est 1<\/span> diversum esse a S(1)=2<\/span>. Hoc est gradus initialis,<\/strong> ubi verificamus proprietatem valere pro primo elemento. <\/p>\n<\/li>\n<li>\nSupponamus proprietatem valere pro quolibet k<\/span>, id est, k\\neq S(k)<\/span>; quod demonstrandum est, est ex hoc sequi etiam S(k)\\neq S(S(k))<\/span>. Hoc est gradus inductivus.<\/strong> Si hi duo gradus complentur, dicitur inductionem esse completam et proprietatem valere pro omnibus naturalibus. <\/p>\n<\/p>\n[1]<\/strong> Ad initium, animadvertamus S(k) \\neq k,<\/span> aequivalere dicere \\neg [k=S(k)]<\/span>.<\/p>\n[2]<\/strong> Sed cum tam k<\/span> quam S(k)<\/span> sint naturales, ex axiomate 2 dicere possumus utrumque successores habere: S(k)<\/span> et S(S(k))<\/span> respective. Ambo quoque naturales sunt.<\/p>\n[3]<\/strong> Deinde, ex axiomate 4 dicere possumus: S(k) = S(S(k))<\/span> implicat k = S(k)<\/span>. Hoc sic scribi potest:<\/p>\n\\left[ S(k) = S(S(k)) \\right] \\rightarrow \\left[k = S(k)\\right]<\/span>\nquod per contrapositi regulam aequivalet dicere:<\/p>\n\\neg \\left[k = S(k)\\right] \\rightarrow \\neg \\left[ S(k) = S(S(k)) \\right] <\/span>\n[4]<\/strong> Denique, applicando modus ponens inter ultimam expressionem et illam quae in gradu [1]<\/strong> obtenta est, habetur:<\/p>\n\\neg \\left[ S(k) = S(S(k)) \\right] <\/span>\nquod idem est ac dicere<\/p>\n S(k) \\neq S(S(k)) <\/span>\nItaque demonstravimus: si constat S(k) \\neq k,<\/span>, tum etiam constat S(k) \\neq S(S(k))<\/span>; cum autem manifestum sit 1\\neq 2<\/span>, inductio est completa et scribi potest:<\/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>Commentarius de demonstrationibus<\/h3>\nQuamquam proprietas enuntiata in exemplo satis manifesta est, in mathematica usitatum est demonstrationes non semper talem evidentiam conservare. Haec demonstratio quam modo vidimus est exemplum eorum quae in labore mathematicae usualiter fiunt. Ad te adiuvandum in comprehensione technicarum deductionis propriae mathematicis, suadeo ut consulas materiales destinatos pro cursu <a href=\"http:\/\/toposuranos.com\/material\/es\/category\/matematica\/logica-matematica\/logica-proposicional\/\" target=\"_blank\" rel=\"noopener\">logicae mathematicae.<\/strong><\/a><\/p>\n<\/body> \n<\/html><\/p>\n","protected":false},"excerpt":{"rendered":"Numeri Naturales et Axiomata Peano Numeri Naturales et Axiomata Peano COMPENDIUM Haec lectio versatur de numeris naturalibus et quomodo per axiomata Peano definiantur: seriem principiorum mathematicorum quae proprietates fundamentales eorum constituunt. Etiam explicat quomodo symbola adhibita sint ad repraesentandos successores numerorum naturalium, quomodo symbolice repraesententur, et usum principii inductionis mathematicae ad probationes inductivas perficiendas. PROPOSITA […]<\/p>\n","protected":false},"author":1,"featured_media":25044,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":1,"footnotes":""},"categories":[1304,1310,1298],"tags":[],"class_list":["post-33295","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-et-geometria","category-algebra-generalis","category-mathematica"],"yoast_head":"\n<title>Numeri Naturales et Axiomata Peano - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"\ud83d\udd0d Explora numeros naturales per axiomata Peano, modum eorum repraesentationis symbolicae, et principium inductionis mathematicae.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/la\/numeri-naturales-et-axiomata-peano\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Numeri Naturales et Axiomata Peano\" \/>\n<meta property=\"og:description\" content=\"\ud83d\udd0d Explora numeros naturales per axiomata Peano, modum eorum repraesentationis symbolicae, et principium inductionis mathematicae.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/la\/numeri-naturales-et-axiomata-peano\/\" \/>\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=\"2021-02-28T00:00:56+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-06-04T21:38:19+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=\"Numeri Naturales et Axiomata Peano\" \/>\n<meta name=\"twitter:description\" content=\"\ud83d\udd0d Explora numeros naturales per axiomata Peano, modum eorum repraesentationis symbolicae, et principium inductionis mathematicae.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/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\":\"https:\/\/toposuranos.com\/material\/la\/numeri-naturales-et-axiomata-peano\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/toposuranos.com\/material\/la\/numeri-naturales-et-axiomata-peano\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Numeri Naturales et Axiomata Peano\",\"datePublished\":\"2021-02-28T00:00:56+00:00\",\"dateModified\":\"2025-06-04T21:38:19+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/toposuranos.com\/material\/la\/numeri-naturales-et-axiomata-peano\/\"},\"wordCount\":767,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"https:\/\/toposuranos.com\/material\/la\/numeri-naturales-et-axiomata-peano\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/numerosnaturales.jpg\",\"articleSection\":[\"Algebra et Geometria\",\"Algebra Generalis\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/toposuranos.com\/material\/la\/numeri-naturales-et-axiomata-peano\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/toposuranos.com\/material\/la\/numeri-naturales-et-axiomata-peano\/\",\"url\":\"https:\/\/toposuranos.com\/material\/la\/numeri-naturales-et-axiomata-peano\/\",\"name\":\"Numeri Naturales et Axiomata Peano - 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