1. ## Unprovable statements

Read up about Godel's proof from Clawson's Math Mysteries. He says that statements exist that can't be proven within a system. What's bothersome is that Godel's proof doesn't indicate how to determine these statements (which is why I don't fully accept his proof).

2. To add I'm a bit like Kronecker when I ask for an example of an unprovable statement.

3. It sounds like you're thinking about decideability: given a proposition, decide whether it is provable from within the system or not. Is that correct? Isn't that is a separate issue from completeness? Godel did exhibit a proposition that was true and yet could not be proven within second-order logic.

4. The book points out that consistency and completeness are different which I agree with. For me it's uncomfortable that a system that at least is based on arithmetic can never be proven to be complete as it'll have an undecidable statement.

While we're on this subject, when Clawson talks about arithmetic, is he referring to Peano's axioms?

5. For me it's uncomfortable that a system that at least is based on arithmetic can never be proven to be complete as it'll have an undecidable statement.
You and a lot of other mathematicians! I think all it says is that we'll never know everything, since there's infinitely many things to learn.

While we're on this subject, when Clawson talks about arithmetic, is he referring to Peano's axioms?
I haven't read Clawson's book, so I'm afraid I don't know. Just looking at the index on Amazon.com, he doesn't seem to refer to Peano's axioms explicitly in the text anywhere, unless that's a missing entry in the index. That's all I can say.