I have a very simple question, that I still haven't found an answer to yet: Gödel is said to have proven that Peano Arithmetic (or any system capable of expressing it) can't prove its own consistency.
His proof relies on the notion that we can construct a statement, that, on a meta-level, means "This statement can not be proven" and from this he follows that the arithmetical statement itself cannot be proven.
But if I see it correctly, this conclusion can't be derived formally. As I see it, his proof shows that the statement that is represented ("This statement can't proven") cannot be proven, but not that the statement it is represented with (an arithmetical statement) can't be proven.
Gödels proof confuses meaning with meta-meaning; he follows from the impossibility of proving the meta-statement the impossibility of proving the actual statement, which is not a provably valid step (though it may be intuitively valid, that is debateable).
So, as most mathematicians would disagree with this, how does Gödel show that the statement that the self-referential unprovability statement is represented with can't be proven?
It seems to me Gödel's theorem can't be proven. That a system can't prove its own consistency is just true because it is simply obvious that no system can prove its own consistency for the very simple reason that the notion of consistency ultimately can't be formalized.
Almost all mathematicians would disagree with me, but what is wrong with my argument that Gödel confuses statement and meta-statement (which might be valid, but can't be proven to be valid)?