{"id":16,"date":"2010-04-28T12:42:16","date_gmt":"2010-04-28T10:42:16","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=16"},"modified":"2010-05-01T03:07:08","modified_gmt":"2010-05-01T01:07:08","slug":"01infinity","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2010\/04\/28\/01infinity\/","title":{"rendered":"{0,1}<sup>\u221e<\/sup>"},"content":{"rendered":"<p style=\"text-align: justify;\">It is amazing to realize how complex things in Mathematics and in  Computer Science can be reduced after all to 0 and 1, in other words, to the  simple notion of difference... Is it beautiful or disappointing? Well, maybe  both! Any resemblance to actual events is coincidental. In fact, and to be  more precise, we must say <span class=\"cite\">sequences of 0 and 1<\/span>,  making then more apparent the role of \u221e. In a way, Computer Science is  the reign of finite sequences of 0 and 1 while Mathematics is the reign of infinite sequences of 0 and 1, in other words, the reign of \u221e. Between the two, you may take a look at the concepts of <a href=\"http:\/\/en.wikipedia.org\/wiki\/Turing_machine\">Turing machines<\/a> and <a href=\"http:\/\/en.wikipedia.org\/wiki\/Kolmogorov_complexity\">Kolmogorov complexity<\/a>. You may also take a look at <a href=\"http:\/\/en.wikipedia.org\/wiki\/Peano_axioms\">Peano axioms<\/a> and <a href=\"http:\/\/en.wikipedia.org\/wiki\/G%C3%B6del%27s_incompleteness_theorems\">G\u00f6del's incompleteness  theorems<\/a>. In Probability Theory, the modelling of the Heads-or-Tails coin-tossing game involves a probability measure on {0,1}<sup>\u221e<\/sup>, the set of infinite sequences of 0 and 1. It has been shown recently that in a sense, almost all large statements  are  indecidable, see for instance<\/p>\n<ul style=\"text-align: justify;\">\n<li><a href=\"\/scripts\/search.php\/?q=Jean-Paul+Delahaye\">Delahaye,  J.-P.<\/a><br \/>\n<strong><span xml:lang=\"fr.FR\">Presque tout est ind\u00e9cidable !<\/span><\/strong><br \/>\n<a title=\"PLS: ISSN 0153-4092 (printed)\" href=\"\/scripts\/search.php\/?q=Pour+la+Science\"><span class=\"sc\">Pour  la Science<\/span><\/a> 375 (January 2009)<\/li>\n<li> <a href=\"\/scripts\/search.php\/?q=Cristian+S+Calude\">Calude, C. S.<\/a> (Editor)<br \/>\n<strong>Randomness and Complexity: From Leibniz to Chaitin<\/strong><br \/>\nWorld Scientific (2007) <a href=\"http:\/\/en.wikipedia.org\/wiki\/International_Standard_Book_Number\"><abbr title=\"International Standard Book Number\">ISBN<\/abbr><\/a> <a title=\"Click to buy a copy of the book on the net\" href=\"\/scripts\/search.php\/?q=Randomness+and+Complexity:+From+Leibniz+to+Chaitin+ISBN+978-9812770820\">978-9812770820<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>It is amazing to realize how complex things in Mathematics and in Computer Science can be reduced after all to 0 and 1, in other&#8230;<\/p>\n<div class=\"more-link-wrapper\"><a class=\"more-link\" href=\"https:\/\/djalil.chafai.net\/blog\/2010\/04\/28\/01infinity\/\">Continue reading<span class=\"screen-reader-text\">{0,1}&lt;sup&gt;\u221e&lt;\/sup&gt;<\/span><\/a><\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":83},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/16"}],"collection":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/comments?post=16"}],"version-history":[{"count":0,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/16\/revisions"}],"wp:attachment":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/media?parent=16"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/categories?post=16"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/tags?post=16"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}