Cantor\u2019s diagonal argument claims that \u211d is uncountable. When we see real numbers as points on the number line, we can put a name on each point and put the names into a list without contravening Cantor\u2019s diagonal argument because we cannot create a diagonal from a list of names. <\/p>\n\n\n\n
However, we do not need such a impossible list, but just to split R into two parts, S2 and S10. the members of S2 are real numbers expressed in binary, those of S10 in decimal. We create a list of real numbers by picking one member\nfrom S2 and one member\nfrom S10 alternately and forever. This list is a composite list whose members are in binary and\ndecimal alternately. The diagonal of\nthis list is a sequence of binary and decimal digits\nalternately and out-of-the-list-number cannot be\nconstructed from it.<\/p>\n\n\n\n
In fact, composite\nlist can be created in splitting R into many subsets in numeral systems of different\nbases from which no\nout-of-the-list-number\ncan be created and there is no real\nnumber excluded from the composite list.\nBecause the composite list is constructed from the whole R and no\nreal number is found outside, the composite list\ncontains R.<\/p>\n\n\n\n
If there is one list that contains R we can already conclude\nthat R is countable.\nBut the permutation\nof the subsets of R can create a huge number of different composite\nlists which all contain R. So, we conclude\nwith confidence that R is countable.\nThen Cantor\u2019s diagonal argument fails.<\/p>\n\n\n\n
Cantor\u2019s\ndiagonal argument expresses real\nnumbers only <\/strong>in one\nnumeral system, which\nrestricts the used list. If a binary list is shown\nnot to contain R, this can be\ncaused either by \u201clist\u201d or by \u201cbinary\u201d. Because Cantor has focused only on \u201clist\u201d overlooking\n\u201cbinary\u201d, this is the flaw that\nbreaks Cantor\u2019s diagonal argument which then does not prove \u211d uncountable.\n<\/p>\n\n\n\n For more detail\nof this\nstudy please read the complete paper here:<\/p>\n\n\n\n \u00abReal numbers and points on the number line<\/a> with regard to Cantor\u2019s diagonal argument<\/a>\u00bb<\/p>\n\n\n\n https:\/\/www.academia.edu\/88279926\/Real_numbers_and_points_on_the_number_line_with_regard_to_Cantors_diagonal_argument<\/a><\/p>\n","protected":false},"excerpt":{"rendered":" Cantor\u2019s diagonal argument claims that \u211d is uncountable. When we see real numbers as points on the number line, we can put a name on each point and put the names into a list without contravening Cantor\u2019s diagonal argument because … Continue reading