From Glick, but about mathematics and Godel

About midway through The Information Glick brings in Godel, yes the Godel of Godel, Escher, Bach who I never knew anything about. Apparently, Whitehead and Russel in Principia Mathematica contended that all could be reduced to mathematics as long as you discarded anything referencing itself (at least I think that's what they said-my ignorance in this area is huge).  Godel, in contrast, said they are wrong. He agreed the math was a science before all others or that everything could be reduced to math (and its this point that interests me) but to finish summarizing Godel. But despite his agreement, Godel discovered that there are truths in math that can NOT be proved. Now, I'm trying to understand Godels thought process. He believed that the symbols used to express mathematical functions were arbitrary-they could be anything and this he would chose to make them all numbers. And, thus, all mathematical formulas could be expressed in one number. Through this simple (but so complex I really can't follow it) process, he demonstrated there were many formulas that could be expressed but could not be proven. He concluded, " within pm and within any consistent logical system capable of elementary arithmetic, there must always be such accursed statements, true but unprovable" (p. 184- although this whole discussion takes place over pp. 180-185). From this, he concluded that numbers can encode all reasoning.

What Turing and Godel concluded together was that all knowing is ultimately self referential and thus cannot know "all". To use hofstaders language, the thing hinges on getting  this halting inspector to predict its one behavior  when looking at itself to predict its wn behavior when looking at itself to predict and on and on (p. 212).