I am looking for a simple explanation/outline of the proof of Gödel's Second Incompleteness Theorem, and I haven't yet been able to find anything that is within my grasp. I'm looking for something like: Kenny's Overview of Hofstadter's Explanation of Gödel's Theorem (for his first theorem) - i.e., not very mathematically rigorous, just giving an overview of the main ideas and the thought process behind the proof.
To me, it seems that the (main ideas of the) proof could be made quite simple:
1.) Gödel's first incompleteness theorem proves that "Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory (Kleene 1967, p. 250)." (wikipedia).
2.) The proof works by asserting that in any "effectively generated theory capable of expressing elementary arithmetic", it is possible to formulate a "Gödel statement" which essential says: "This statement is not provable"; this must be a true statement, for if it were false, then it could be proven, and that would lead to contradictions. So we have a true statement which is not provable within the theory. QED.
3.) Now, IF we were able to prove the "Gödel statement", our theory would be inconsistent. Thus, a statement which asserts the consistency of our theory must prove/imply that we CANNOT prove the Gödel statement, or, in other words, it must prove the statement that the Gödel statement is not provable.
4.) BUT - the statement: "The Gödel statement is not provable" IS the Gödel statement. So, in proving its own consistency, a theory must prove the Gödel statement, implying that it is inconsistent.
Nothing that I have read has been anywhere near this simple. What am I missing? Is it really much more complicated that what I have put here?
DISCLAIMER: I am not a mathematician, and I haven't had much in the way of formal mathematics instruction. I am just an interested layperson.