[donh]

A formal system is complete when all its theorems can be demonstrated to be valid formulas according to the rules of the formal system.

I presume you mean that all of its true theorems can be proved.

Godel destroyed the presumption of reducibility forever.

By reducibility, I presume you mean the same as completeness above. Godel's theorem applies to discrete systems only; you cannot make use of the enumeration step of the proof if your domain of discourse is continuous--you have nothing to map to. Amusingly enough, considering Russell's initial assumptions, formal continuous systems, such as plane geometry and The Calculus, are presently thought to be complete, or reducible, if you like, at least in theory. Also, Godel's proof is just a proof, which is a man-made object--it is not a Transcendental Truth, and important proofs that were venerated for years have fallen on their faces before this.

[beaver fever]

" Godel destroyed the presumption of reducibility forever."

By reducibility Godel meant the rules of arithmetic in addition to two truth values plus two logical operators, (addition and subtraction) should obey the conditions of completeness and consistency of formal logic.

His proof proved otherwise. Even for simple arithmetic that only recognized addition and subtraction as logical operators some valid formulas could not be identified as theorems and existing theorems could not be determined to be valid formulas according to the rules of arithmetic.

Quoted from you

"I presume you mean that all of its true theorems can be proved."

Exactly the opposite.

Godel proved that any logic that contained quantifiers, (For some, For Every) would be incomplete and inconsistent.

Godels proof is so powerful that it blew the bottom out of mathematical logic and consigned to a sidebar in the history of Philosophy.

If logic can't construct a coherent account of the underlying rules of addition and subtraction then it can't account for anything in the world. (Paraphrasing Wittgenstein)