CANTOR'S PARADISE Painting

In other works, we conveyed the idea that one can enumerate all the fractions. The natural next step is to ask whether one can do the same for the real numbers, i.e. all the fractions along with the irrational numbers. Cantor proved in 1891 that this is not possible, and he did so using an ingenious diagonal construction, which is represented in the work. 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.

Painting:Oil on Canvas

Original:One-of-a-kind Artwork

Size:19.7 W x 27.6 H x 0.7 D in

Delivery Time:Typically 5-7 business days for domestic shipments, 10-14 business days for international shipments.