know it all » (say) no (to) it all » no, no, no ! » nien, nien, nien ! » 999 !


In 1927 the French mathematician Émile Borel 1,2,3 (1871-1956) gave a method for constructing a real number that he dubbed the “know it all number”.  This real number encodes the answer to all possible yes/no questions!


Borel’s “know it all number” is the motivation for Jorge Luis Borges’ 4 famous short story The Library of Babel  5,6.


The details of Borel’s construction are given in the fascinating and very readable article “How Real are Real Numbers?” 7 by Gregory Chaitin 8. (This article can be downloaded from Chaitin’s website.) 9  In this article Chaitin nimbly weaves together a story on “nameable” numbers, Turing’s halting problem and Chaitin’s Omega 10,11.


The spirit of Chaitin’s question “How real are the reals?” is captured in the following Chaitin quote.


Geometrically a real number is the most straightforward thing in the world, it's just a point on a line. That's quite natural and intuitive. But arithmetically, that's another matter. The situation is quite different. From an arithmetical point of view reals are extremely problematical, they are fraught with difficulties! 12









7.    Chaitin, G., “How Real are Real Numbers?”, International Journal of

         Bifurcation and Chaos, Vol. 16, No. 6 (2006) 1841-1848.





12.   Chaitin, G.,”Epistemology as Information Theory: From Leibniz to Ω”,

       Collapse, Vol. 1, 2006 27-51.