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.

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.