BOULEVARD DU TEMPLE III Painting

This is a representation of Cantor's 1891 proof that this it is not possible to count the reals. Cantor proved this using an ingenious diagonal construction. One can proceed by Reductio Ad Absurdum and show that from any assumed enumeration (of just the real numbers between 0 and 1), at least one number must be missing, which can be constructed by taking any number that is different from the I’th number in the enumeration in the I’th digit. Today Cantor’s results are recognized as revolutionary and classical, but they were initially considered controversial, and he was even accused of humbug. One reason was that his theory was associated with a number of paradoxes. The perhaps best known paradox was Russell’s paradox. Frege had tried to formalize Cantor’s set theory in a formal system, but Russel discovered in 1901 that there was a paradox hidden in Frege’s formalization. Russell considered the set of all sets that do contain themselves and then asked the question whether this set contains itself. If the answer is yes, it is no, and vice versa – i.e. a blatant paradox. This work is part of the series THEOREMS AND PROOFS. The works illustrate – directly or indirectly – a theorem with a beautiful proof, according to the criteria put forth by the mathematician Hardy in his classical book A Mathematician's Apology: surprise combined with something inevitable, economy and depth.