I’m scared. Some appears to have proven that the Collatz 3n+1 Conjecture is unprovable. The reason that this result scares me is because it implies that the Collatz 3n+1 Conjecture is true, since if there exists a number n for which there is no m with T^(m)(n)=1, then we can clearly disprove it. I don’t think it should be possible to “prove” statements true using this logic. How can it be verified that a statement is true by showing that it cannot be proven? People have given this argument for the Riemann Hypothesis’s truth if it turns out to be formally undecidable for quite a while, but I don’t know how many people think that such logic can really be used to decide a problem.
On a lighter note, I bought myself a wonderful book yesterday. It’s called Counterexamples in Analysis by Bernard Gelbaum and John Olmsted. It basically goes through all sorts of things that can go wrong in analysis. Do similar books exist for other branches of mathematics?