{"id":7822,"date":"2026-05-27T22:01:58","date_gmt":"2026-05-27T19:01:58","guid":{"rendered":"https:\/\/www.schooler.org.ua\/uk-uamatematichnij-dokaz-u-bud-jakij-sistemi-ye-dirki-ru\/"},"modified":"2026-05-27T22:01:58","modified_gmt":"2026-05-27T19:01:58","slug":"uk-uamatematichnij-dokaz-u-bud-jakij-sistemi-ye-dirki-ru","status":"publish","type":"post","link":"https:\/\/www.schooler.org.ua\/cs\/uk-uamatematichnij-dokaz-u-bud-jakij-sistemi-ye-dirki-ru\/","title":{"rendered":"Matematick\u00fd d\u016fkaz: ka\u017ed\u00fd syst\u00e9m m\u00e1 d\u00edry"},"content":{"rendered":"

Kolegov\u00e9 m\u011b \u010dasto \u017e\u00e1daj\u00ed o pomoc s \u010d\u00edsly. Mysl\u00ed si, \u017ee um\u00edm matematiku. Ironi\u00ed je, \u017ee jsem hrozn\u00e1 v po\u010d\u00edt\u00e1n\u00ed v hlav\u011b. <\/p>\n

Lid\u00e9 \u010dasto \u0161patn\u011b rozum\u011bj\u00ed matematice. Mysl\u00ed si, \u017ee to d\u011bl\u00e1 rychl\u00e9 s\u010d\u00edt\u00e1n\u00ed a od\u010d\u00edt\u00e1n\u00ed ve va\u0161\u00ed hlav\u011b, zat\u00edmco si kupujete k\u00e1vu. Ne. Skute\u010dn\u00e1 matematika je struktura sv\u011bt\u016f<\/em>. <\/p>\n

M\u00fdtus nadace<\/h3>\n

V\u0161e za\u010d\u00edn\u00e1 pravidly. Axiom. Z\u00e1kladn\u00ed pravdy, kter\u00e9 p\u0159ijmete, aby hra mohla za\u010d\u00edt. Na tomto z\u00e1kladu postav\u00edte v\u0161e ostatn\u00ed: mno\u017einy se stanou \u010d\u00edsly, \u010d\u00edsla se stanou funkcemi a funkce se p\u0159em\u011bn\u00ed v geometrii. To v\u0161e je zalo\u017eeno na stejn\u00e9m po\u010d\u00e1te\u010dn\u00edm podla\u017e\u00ed. <\/p>\n

Matematici hr\u00e1li dlouhou dobu nebezpe\u010dnou hru s rovnov\u00e1hou. Cht\u011bli pou\u017e\u00edt co nejm\u00e9n\u011b pravidel, ale st\u00e1le dost na to, aby popsali modern\u00ed vesm\u00edr. A tato pravidla se musela zd\u00e1t spr\u00e1vn\u00e1<\/em>. Intuitivn\u011b srozumiteln\u00e9. Stejn\u011b jako v\u00fdrok \u201eexistuje pr\u00e1zdn\u00e1 mno\u017eina\u201c. Prost\u011b to d\u00e1v\u00e1 smysl. <\/p>\n

V roce 1900 se v\u0161ichni shodli na syst\u00e9mu ZFC. Zermelo-Frenkelova teorie mno\u017ein s axiomem v\u00fdb\u011bru. Dev\u011bt pravidel. V\u0161e. Tohle byl z\u00e1klad. <\/p>\n

Nebo si to alespo\u0148 mysleli. <\/p>\n

G\u00f6del ni\u010d\u00ed v\u0161echno<\/h3>\n

Matematici milovali sv\u016fj z\u00e1klad. Snili o syst\u00e9mu, kter\u00fd by m\u011bl dv\u011b kvality z\u00e1rove\u0148:<\/p>\n

    \n
  1. \u00daplnost.<\/strong> Jak\u00e1koli pravda m\u016f\u017ee b\u00fdt prok\u00e1z\u00e1na. <\/li>\n
  2. Konzistence.<\/strong> Rozpory jsou nep\u0159ijateln\u00e9. <\/li>\n<\/ol>\n

    P\u0159ich\u00e1z\u00ed rok 1931. Na jevi\u0161ti – Kurt G\u00f6del. Je mu p\u011btadvacet let. Shod\u00ed bombu, kter\u00e1 rozbije samotn\u00fd z\u00e1klad. <\/p>\n

    Jeho prvn\u00ed v\u011bta o ne\u00faplnosti je nemilosrdn\u00e1. Uv\u00e1d\u00ed, \u017ee v ka\u017ed\u00e9m dostate\u010dn\u011b siln\u00e9m a konzistentn\u00edm syst\u00e9mu existuj\u00ed tvrzen\u00ed, kter\u00e1 nelze prok\u00e1zat ani pravdiv\u011b, ani nepravdiv\u011b. Te\u010dka. A pak p\u0159ich\u00e1z\u00ed druh\u00e1 v\u011bta. Je\u0161t\u011b hor\u0161\u00ed. Syst\u00e9m nem\u016f\u017ee prok\u00e1zat svou vlastn\u00ed konzistenci. <\/p>\n

    Zn\u00ed to akademicky, samoz\u0159ejm\u011b. Abstraktn\u00ed logika. G\u00f6delovi kolegov\u00e9 doufali, \u017ee jde jen o podivnost. N\u011bjak\u00e1 teoretick\u00e1 chyba, postr\u00e1daj\u00edc\u00ed praktick\u00fd v\u00fdznam. M\u00fdlili se. <\/p>\n

    \n

    G\u00f6del dok\u00e1zal, \u017ee jistota m\u00e1 sv\u00e9 hranice. Nem\u016f\u017eete v\u011bd\u011bt v\u0161echno, dokonce ani v r\u00e1mci pravidel, kter\u00e1 jste si sami vytvo\u0159ili. <\/p>\n<\/blockquote>\n

    Vezm\u011bme stejn\u00fd syst\u00e9m ZFC. Je pln\u00e1 neprokazateln\u00fdch tvrzen\u00ed. Nejzn\u00e1m\u011bj\u0161\u00ed je hypot\u00e9za kontinua. Existuje nekone\u010dno mezi cel\u00fdmi \u010d\u00edsly a re\u00e1ln\u00fdmi \u010d\u00edsly? Nev\u00edme. A s na\u0161imi sou\u010dasn\u00fdmi n\u00e1stroji to nikdy nebudeme schopni dok\u00e1zat. Tahle ot\u00e1zka prost\u011b… z\u016fstala viset ve vzduchu. <\/p>\n

    Nevy\u0159e\u0161en\u00fd. Ani ne\u0159e\u0161iteln\u00e9. <\/p>\n

    Tak\u017ee ano, m\u016f\u017eete stav\u011bt sv\u011bty z axiom\u016f. M\u016f\u017eete p\u0159ej\u00edt od jednoduch\u00fdch mno\u017ein ke komplexn\u00ed topologii. Ale ve struktu\u0159e jsou d\u00edry. Vestav\u011bn\u00e1 slep\u00e1 m\u00edsta, kde se pravda skr\u00fdv\u00e1, nav\u017edy mimo dosah. <\/p>\n

    Pro\u010d mysl tou\u017e\u00ed po \u00faplnosti, kdy\u017e je \u00faplnost nemo\u017en\u00e1?<\/p>\n","protected":false},"excerpt":{"rendered":"

    Kolegov\u00e9 m\u011b \u010dasto \u017e\u00e1daj\u00ed o pomoc s \u010d\u00edsly. Mysl\u00ed si, \u017ee um\u00edm matematiku. Ironi\u00ed je, \u017ee jsem hrozn\u00e1 v po\u010d\u00edt\u00e1n\u00ed v hlav\u011b. Lid\u00e9 \u010dasto \u0161patn\u011b rozum\u011bj\u00ed matematice. Mysl\u00ed si, \u017ee to d\u011bl\u00e1 rychl\u00e9 s\u010d\u00edt\u00e1n\u00ed a od\u010d\u00edt\u00e1n\u00ed ve va\u0161\u00ed hlav\u011b, zat\u00edmco si kupujete k\u00e1vu. Ne. Skute\u010dn\u00e1 matematika je struktura sv\u011bt\u016f. M\u00fdtus nadace V\u0161e za\u010d\u00edn\u00e1 pravidly. Axiom. […]<\/p>\n","protected":false},"author":1,"featured_media":7821,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"tdm_status":"","tdm_grid_status":""},"categories":[1],"tags":[],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/www.schooler.org.ua\/cs\/wp-json\/wp\/v2\/posts\/7822"}],"collection":[{"href":"https:\/\/www.schooler.org.ua\/cs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.schooler.org.ua\/cs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.schooler.org.ua\/cs\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.schooler.org.ua\/cs\/wp-json\/wp\/v2\/comments?post=7822"}],"version-history":[{"count":0,"href":"https:\/\/www.schooler.org.ua\/cs\/wp-json\/wp\/v2\/posts\/7822\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.schooler.org.ua\/cs\/wp-json\/wp\/v2\/media\/7821"}],"wp:attachment":[{"href":"https:\/\/www.schooler.org.ua\/cs\/wp-json\/wp\/v2\/media?parent=7822"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.schooler.org.ua\/cs\/wp-json\/wp\/v2\/categories?post=7822"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.schooler.org.ua\/cs\/wp-json\/wp\/v2\/tags?post=7822"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}