I recently wrote the outline of the proof of Gödel’s incompleteness theorems for Simple Wikipedia. I would like to get feedback on its clarity and logical validity so that I can make further improvements. I’ve reproduced relevant sections from the Simple Wikipedia article below.
Gödel’s Theorems
Gödel said that every non-trivial formal system (consistent and axiomatic system with theorems listable by following an algorithm) is incomplete and not provably consistent:
There will always be questions that cannot be answered, using a certain set of axioms; there are truths that cannot be proved using the axioms of the system.
You cannot prove that a system of axioms is consistent according to the axioms of the system.
Outline of Proof of Gödel’s Theorems
The proof of Gödel’s theorems is based on forming and considering special self-referential statements.
One way to prove the theorems is to consider the first theorem before considering the second theorem. There are three parts in this proof:
A. Consider the first theorem and the truth of this first statement “This statement cannot be proved”: proving this first statement would be a contradiction because there would then be a proof for a statement that cannot be proved. For the sake of the consistency of the system, the first statement and hence the first theorem are therefore true but unprovable.
B. Recall from part A that if we were able to prove the first statement then our system would be inconsistent. Thus to prove the consistency of the system, we must prove that the first statement is not provable (ie: prove the contrapositive of the claim “if we were able to prove the first statement then our system would be inconsistent”). Because of self-reference in the first statement “this statement cannot be proved” (ie: proving the first statement is not provable is proving the first statement), therefore proof of the consistency of the system requires proof of the first statement.
C. Recall in part A that consistency of the system is why the first statement is true and unprovable. Hence by the truth of the first statement and the conclusion in part B, it follows that the consistency of the system cannot be proved with the axioms of the system. Hence the second theorem is established. The truth of the first theorem is established in part A.
Thanks for taking the time to proof-read and contemplate the statements!