A concise summary of the proof of Goedel's Theorem and some related Paradoxes.

Another proof of Goedel's Theorem, from a computer science perspective.

A demonstration of the truth of Goedel's Theorem can be found here. (See the quote by Rucker contained therein.)

Another excellent discussion of Goedel's Incompleteness Theorem.

A more technical discussion of the link between Goedel's theorem and Cantor's diagonlization can be found here

Here is a description, in words, of Goedel's proof: (taken from here)

Here is a simpler proof that no number system can generate all the statements which might be true within it. This proof is based on the writings of A. W. Moore and Roger Penrose.

#1. POINT TO PROVE: IT IS IMPOSSIBLE TO DERIVE ALL MATHEMATICAL TRUTH FROM ANY SET OF SELF-EVIDENT AXIOMS. #2. IF ALL MATHEMATICAL TRUTHS CAN BE DERIVED FROM A CHOSEN SET OF AXIOMS, THEN, IN PRINCIPLE, AN ALGORITHM "A" CAN BE CREATED TO TEST WHETHER OR NOT ANY GIVEN THEOREM DERIVES FROM THE CHOSEN AXIOMS, I.E.: WHETHER OR NOT IT IS TRUE OR FALSE. #3. AT PRESENT WE DO NOT HAVE SUCH AN ALGORITHM. IF A CAN BE SHOWN TO BE IMPOSSIBLE, THEN #1 IS ESTABLISHED. #4. LIST THE FACTUAL STATEMENTS WHICH CAN BE MADE ABOUT NUMBERS. EXAMPLES OF SUCH STATEMENTS ARE "X IS EVEN," "X IS ODD," "X IS PRIME","X IS LESS THAN 100," ETC. #5. CREATE A TABLE OF SUCH STATEMENTS, BEGINNING WITH THE SIMPLEST AND MOVING TO THE MORE COMPLEX. WE WILL CALL OUR STATEMENTS 1, 2, 3, 4... NOW WE NOTE THAT OUR TABLE CAN REFER TO ITS OWN STATEMENTS. SUPPOSE STATEMENT 0 MEANS: "X IS EVEN", STATEMENT 1 "X IS ODD" ETC... WE LET THE VERTICAL AXIS REPRESENTS THE STATEMENT NUMBER. THE HORIZONTAL AXIS REPRESENTS ALL NUMBERS FROM 0 TO INFINITY. WE THEN ASK OURSELVES FOR EACH NUMBER IN THE HORIZONTAL AXIS, "IS THE VERTICAL STATEMENT TRUE OF THIS NUMBER?" WE WRITE Y BELOW IT IF IT IS TRUE, AND N IF IT ISN'T: 0 1 2 3 .... 0 (EVEN) N N Y N... 1 (ODD) N N Y Y 2 (PRIME) N N Y Y... 3 (x<100 ) Y Y Y Y.... ... .................. #6. FOR ANY NATURAL NUMBER (HORIZONTAL LINE) WE NOW HAVE A METHOD OF DECIDING IF THE VERTICAL STATEMENT IS TRUE. SINCE EVERY POSSIBLE STATEMENT OF THE SYSTEM CAN APPARENTLY BE LISTED AND SINCE EVERY NATURAL NUMBER CAN ALSO BE LISTED, IT APPEARS WE HAVE A COMPLETE SYSTEM OF NATURAL NUMBERS AND AXIOMS. NOTICE THAT EACH STATEMENT ON THE VERTICAL AXIS PRODUCES ITS OWN UNIQUE HORIZONTAL LINE OF Ys AND Ns. #7. CREATE A NEW WELL-DEFINED SEQUENCE OF Ys AND Ns BY FOLLOWING A DIAGONAL ON THE CHART WE HAVE JUST CREATED. DO THIS BY TURNING EACH DIAGONAL ELEMENT INTO ITS OPPOSITE. THE N AT 0/0 ON THE TABLE BECOMES A Y. THE Y AT 1/1 BECOMES AN N. THE Y AT 2/2 BECOMES AN N. THE Y AT 3/3 BECOMES AN N AND SO FORTH. WE GET YNNN... DOES ANY STATEMENT WHICH HAS ALREADY BEEN GIVEN PRODUCE THIS NEW SEQUENCE? #8. STATEMENT 0 DOESN'T BECAUSE IT HAS AN N WHERE THE NEW STATEMENT HAS Y. 1, 2, AND 3 DON'T BECAUSE THEY HAVE Ys WHERE THE NEW STATEMENT HAS Ns. THIS WOULD HOLD TRUE TO INFINITY IF WE COULD MAKE OUR TABLE THAT LONG, #9. WE KNOW WE LEGITIMATELY CREATED THIS NEW Y & N PATTERN, IE: IT IS TRUE. YET NONE OF THE EXISTING AXIOM STATEMENTS PRODUCE THIS DIAGONAL STATEMENT. A NEW AXIOM IS NEEDED TO EXPRESS THE DIAGONAL. 10. IF WE WRITE A NEW STATEMENT (CALL IT R) THAT INCLUDES A PROCEDURE FOR MAKING THIS DIAGONAL , AT SPACE R/R A NEW DIAGONAL LETTER WILL APPEAR AND WE WILL HAVE TO ADD STATEMENT S TO REPRESENT THIS NEW SEQUENCE. BUT AT S/S A NEW DIAGONAL NUMBER WILL APPEAR, REQUIRING A STATEMENT T AND SO ON, INFINITELY. 11. THEREFORE ALGORITHM A IS IMPOSSIBLE, WHICH IS THE PROOF REQUIRED BY #2. IT IS IMPOSSIBLE TO AUTOMATICALLY DERIVE ALL POSSIBLE MATHEMATICAL TRUTH.