Thoughts on #10:

I am confused about this exercise. The standard/modern proof of Gödel’s second incompleteness theorem uses the Hilbert–Bernays–Löb derivability conditions, which are stated as (a), (b), (c) in exercise #11. If the exercises are meant to be solved in sequence, this seems to imply that #10 is solvable without using the derivability conditions. I tried doing this for a while without getting anywhere.

Maybe another way to state my confusion is that I’m pretty sure that up to exercise #10, nothing that distinguishes Peano arithmetic from Robinson arithmetic has been introduced (it is only with the introduction of the derivability conditions in #11 that this difference becomes apparent). It looks like there is a version of the second incompleteness theorem for Robinson arithmetic, but the paper says “The proof is by the construction of a nonstandard model in which this formula [i.e. formula expressing consistency] is false”, so I’m guessing this proof won’t work for Peano arithmetic.

Eliezer Yudkowsky has made similar points in e.g. “Unteachable Excellence” (“much of the most important information we can learn from history is about how to not lose, rather than how to win”, “It’s easier to

avoidduplicating spectacular failures than to duplicate spectacular successes. And it’s often easier to generalize failure between domains.“) and “Teaching the Unteachable”.