Godelの証明

最近Godel's　Proofと言う本を再読しています。Godelは二つの理論で数学界に知られているそうです。
It seems that he formulated both Incompleteness and Completeness theorems. The completeness system states that if a formula is logically valid then a finite proof of the formula exists. A formula is considered to be logically valid if it is true in every structure for its language. His incompleteness theorem states that if any sufficiently strong (?) theory of arithmetic is consistent, then there exists a formula which can neither be proven nor disproven within the theory. It seems that this would make the theory inconsistent.
The text I am reading now seems to be related to the incompleteness theorem, as it deals with a refutation of the application of the axiomatic method to various branches of mathematics. The axiomatic method, of which Euclidean geometry is an oft-used example, enables all true statements about the usiverse of discourse to be derived from a finite number of axioms which are accepted as true. Russell and Whitehead, in their Principia Mathematica, attempted to ground all of mathematics in logic, thus applying the axiomatic method to all branches, it seems. Godel demonstrated the impossibility of this by showing that it is impossible to show the consistency of any deductive system unless one adopts priniciples so compex that their own consistency is in doubt.
Apparently, these conclusions were cause for a serious reexamination of both mathematics and philosophy, but I have yet to determine what the exact influence was.
One of the fundamental problems approached by mathematicians in the nineteenth century was one of consistency: whether a system of statements is such that no mutually contradictory theorems can be deduced from it. This is apparently equated to the fact that all the statements can be proven to be true for all applications within their domain. An example of a contradiction, or "antinomy", was given by Russell. Classes are either normal or not normal. Define a normal class as one which does not contain itself (eg mathematicians). Then let N be the set of all normal classes. If N contains itself, then it is normal (as all normal classes are in N), but it cannot be normal (as normal defined as class that does not contain self). If N does not contain itself, then N is not normal (as all normal classes are in N), but then N is normal (by definition of normal). So in either case N is normal and non-normal.