I do a detailed analysis of Cantor\u2019s theory of uncountable sets. The logic of his proofs has some weaknesses. I propose an axiom and a solution to continuum hypothesis.<\/p>\n\n\n\n
The main idea is:
\nAssumption of Cantor\u2019s proofs: All real numbers (set R) are in a list (list L).<\/p>\n\n\n\n
This assumption means R=L, considering L as a set. This makes the claim \u201ca real number is created but is not in the list L\u201d wrong. Indeed, if a number is outside L, it is outside R too. So, the statement \u201cthe created real number is not in the list L\u201d means it is not in R and is not a real number, which is equivalent to claim that a real number is not real number. This is absurd but Cantor\u2019s diagonal argument and nested intervals proof both claim that a real number is not in the list L and thus, is not a real number, which make them wrong.<\/p>\n\n\n\n
On the other hand, Cantor’s both proofs search for contradiction. Can \u201cthis real number is not a real number\u201d be the contradiction? No. The contradiction of the proofs is in the third step: sout \u201cis not in the list\u201d. By failing to create sout the second step collapses before the third step declares the contradiction.<\/p>\n\n\n\n
See the paper <\/p>\n","protected":false},"excerpt":{"rendered":" I do a detailed analysis of Cantor\u2019s theory of uncountable sets. The logic of his proofs has some weaknesses. I propose an axiom and a solution to continuum hypothesis. The main idea is: Assumption of Cantor\u2019s proofs: All real numbers … Continue reading
https:\/\/www.academia.edu\/86410224\/Examination_of_Cantors_proofs_for_uncountability_and_axiom_for_counting_infinite_sets<\/a><\/p>\n\n\n\n