Identifying Cantor’s Diagonal Argument as an Antinomy: Exploring Complementary Analysis Techniques

It has been noted that self-referential and ambiguous definition formulas are accompanied by complementary self-referential antinomy formulas, which give rise to contradictions. This made it possible to reexamine the ancient antinomies, Cantor’s Diagonal Argument (CDA), and the method of nested intervals, which is the basis for evaluating the existence of uncountable sets. CDA is seen by many mathematicians as a beautiful and easy argument whose consequences lead to different powers of infinity, thus opening the door to mathematical paradise. The simple reasoning he uses seems to be a highly effective way of defining a sequence, implying a proof of uncountability because the diagonals of a two-character list turn into opposites, and at first glance, there is nothing to disprove the argument. A new look at the complementarity of Cantor’s formulas in this article refreshes the hunches of Wittgenstein and other opponents of the existence of uncountable sets, putting CDA in a different light. In Cantor’s theorem, a formula was used to define a set that cannot be the value of any argument of any function f: ℕ → P(ℕ). Examining the complement of the created set, we find that this complement must be unique due to the bijective reversal 0 ↔ 1 of the signs of the indicator function of the Cantor set. However, at the same time, its definition generates two different sets for one argument, which contradicts the basic property of every function. Other studies confirm the invalidity of Cantor’s proofs and the nonexistence of uncountable sets.