In a book by this name, Gregory J. Chaitin (Springer, Singapore, 1999) probes the limits of mathematics. When George Cantor formulated his theory of transfinite sets (whole different orders of infinities, each including the previous and continuing on to infinity), this when analyzed by Bertrand Russell led the latter to certain paradoxes.
The main one was “Is the set of all sets that are not members of themselves a member of the set?” The answer is both yes and no, or rather, if yes then no, and if no then yes. This is quite like the second paradox, that of the Cretan liar: the Cretan said “All Cretans are liars”. If it is true, then it is false; if it is true, then it is false. More tersely, this can be put in a shorter statement, “This sentence is false.” (The one about the barber shaving himself, however, is resolved if the barber is a woman.)
These paradoxes are due to a property called self-reference; this was illustrated copiously and ingeniously in Hofstadter’s “Gödel, Escher, Bach”). Russell proceeded to eliminate the paradox from formal logic, declaring it inadmissible. However, this seems rather arbitrary, like ruling out the existence of a creature that does not conform to your ideas of biology; yet it exists. (Like proving by engineering principles that a bumblebee could not fly.) We have since learned how to deal with it by inventing “fuzzy logic”, defining the Cretan liar statement “half-true and half/false.
Incidentally, physical phenomena of this nature also exist, e.g. the electric buzzer: is the electric circuit open or closed? Well, half and half, vibrating between the two states. And where are the double bonds in a benzene ring? Vibrating very rapidly between two configurations, actually stabilizing the structure. Of course, quantum theory abounds with fuzzy overlapping states.
Self-reference is actually an instance of intransitivity in a binary system, when you think about it. Usually intransitivity requires at least three members: A is greater than B. B is greater than C, but C is greater than A. Self-reference seems to be an instance of A is more true than B, but B is more true than A. Nature in physical systems deals with this paradox by vibrating the alternative states. But in strict formal logic and mathematics this is impossible, because these are static structures — trying to be eternal, in a way.
Yet Russell should not have ruled out the self-reference paradoxes. It is better to retain them, not deny them, and admit that any formal mathematical system is either in-complete or contradictory (i.e. inconsistent) . This is what Kurt Gödel did in his famous proof; he used a self-referential statement in the proof. We could “vibrate” “incomplete” and “contradictory” if we wished, but “incomplete” seems more acceptable to most mathematicians and ordinary people. This is because, if contradictory statements are admitted into symbolic logic (i.e. if A and not-A are both true), the consequence is that ANYTHING could be proved.
Gödel’s proof of incompleteness defeated Hilbert’s project of establishing a complete formalization of all of mathematics as a way of removing all uncertainties. It meant that there are true statements that are not theorems, i.e. cannot be deduced from a small basic set of axioms, as well as false statements that cannot be proved to be false. (The picture is one of mutually interpenetrating fractal trees of black and white.) Yet the question still remains, how many of these unprovable statements are there? How serious is the contamination? If the exceptions are only self-referential statements, perhaps this is not too serious. We might be able to ignore them, or rule them out, as Russell did.
Alas, it is not so. The malignancy has translocated to other sites, like the metastasis of cancer cells.