I saw this question just after posting an answer of sorts in the preceding discussion:

Knuth's intuition that Goldbach might be unprovable

As far as I know, the "rapidly growing function" method is not fundamentally different from the "statement allows the theory to construct model of itself" technique. It is an interpretation of the latter method, originally provided by Ketonen and Solovay as a more transparent way of understanding the Paris-Harrington proof (which showed that from the finitary function implied by their variant of Ramsey's theorem, one could construct a model of Peano Arithmetic, and this construction could itself be carried out in PA).

To find a model of PA (or ZF, or other expressive formal system) within a given problem such as Goldbach, it has to support the combinatorics entailed in those formal systems, such as the proof-theoretic ordinals up to $\epsilon_0$ or $\Gamma_0$, or large cardinal combinatorics as in Friedman's work. Theorems of Paris-Harrington, Friedman and others that are unprovable in specific systems refer to structures that are reverse-engineered from the combinatorics appearing in mathematical logic.

Thus, to get PA-unprovability from Goldbach or Riemann conjectures one would have to somehow relate prime number distribution to the extremely rigid combinatorics of either (1) models of PA, (2) the syntax of PA as a formal proof system, or (3) $\epsilon_0$ and the proof-theoretic ordinal analysis of PA. This happens only if things like the Weil conjectures and Riemann hypothesis are the tiniest tip of a structural iceberg in number theory, OR, alternatively, if provable relations to formal systems appear generically in lots of problems all over mathematics, in which case there is a machine for resolving large numbers of open problems (e.g., in ZF) en route to demonstrating their unprovability in PA.

EDIT: regarding the discussion below of rapidly growing functions not seeming to appear in Goldbach, it's not necessary that the/a/any Goldbach function $g(n)$ implicit in the conjecture be fast-growing, only that Peano Arithmetic be able to construct from $g(n)$ something else that is rapid. For example, if you look at the set of $n$ such that $g(n) = 3$, this is a subsequence of something analogous to the twin primes, and conceivably could grow very fast if PA can't prove the Twin Prime Conjecture or its ilk.

I saw this question just after posting an answer of sorts in the preceding discussion:

Knuth's intuition that Goldbach might be unprovable

As far as I know, the "rapidly growing function" method is not fundamentally different from the "statement allows the theory to construct model of itself" technique. It is an interpretation of the latter method, originally provided by Ketonen and Solovay as a more transparent way of understanding the Paris-Harrington proof (which showed that from the finitary function implied by their variant of Ramsey's theorem, one could construct a model of Peano Arithmetic, and this construction could itself be carried out in PA).

To find a model of PA (or ZF, or other expressive formal system) within a given problem such as Goldbach, it has to support the combinatorics entailed in those formal systems, such as the proof-theoretic ordinals up to $\epsilon_0$ or $\Gamma_0$, or large cardinal combinatorics as in Friedman's work. Theorems of Paris-Harrington, Friedman and others that are unprovable in specific systems refer to structures that are reverse-engineered from the combinatorics appearing in mathematical logic.

Thus, to get PA-unprovability from Goldbach or Riemann conjectures one would have to somehow relate prime number distribution to the extremely rigid combinatorics of either (1) models of PA, (2) the syntax of PA as a formal proof system, or (3) $\epsilon_0$ and the proof-theoretic ordinal analysis of PA. This happens only if things like the Weil conjectures and Riemann hypothesis are the tiniest tip of a structural iceberg in number theory, OR, alternatively, if provable relations to formal systems appear generically in lots of problems all over mathematics, in which case there is a machine for resolving large numbers of open problems (e.g., in ZF) en route to demonstrating their unprovability in PA.

EDIT: regarding the discussion below of rapidly growing functions not seeming to appear in Goldbach, it's not necessary that the/a/any Goldbach function $g(n)$ implicit in the conjecture be fast-growing, only that Peano Arithmetic be able to construct from $g(n)$ something else that is rapid. For example, if you look at the set of $n$ such that $g(n) = 3$, this is a subsequence of something analogous to the twin primes, and conceivably could grow very fast if PA can't prove the Twin Prime Conjecture or its ilk.

I saw this question just after posting an answer of sorts in the preceding discussion:

Knuth's intuition that Goldbach might be unprovable

As far as I know, the "rapidly growing function" method is not fundamentally different from the "statement allows the theory to construct model of itself" technique. It is an interpretation of the latter method, originally provided by Ketonen and Solovay as a more transparent way of understanding the Paris-Harrington proof (which showed that from the finitary function implied by their variant of Ramsey's theorem, one could construct a model of Peano Arithmetic, and this construction could itself be carried out in PA).

To find a model of PA (or ZF, or other expressive formal system) within a given problem such as Goldbach, it has to support the combinatorics entailed in those formal systems, such as the proof-theoretic ordinals up to $\epsilon_0$ or $\Gamma_0$, or large cardinal combinatorics as in Friedman's work. Theorems of Paris-Harrington, Friedman and others that are unprovable in specific systems refer to structures that are reverse-engineered from the combinatorics appearing in mathematical logic.

Thus, to get PA-unprovability from Goldbach or Riemann conjectures one would have to somehow relate prime number distribution to the extremely rigid combinatorics of either (1) models of PA, (2) the syntax of PA as a formal proof system, or (3) $\epsilon_0$ and the proof-theoretic ordinal analysis of PA. This happens only if things like the Weil conjectures and Riemann hypothesis are the tiniest tip of a structural iceberg in number theory, OR, alternatively, if provable relations to formal systems appear generically in lots of problems all over mathematics, in which case there is a machine for resolving large numbers of open problems (e.g., in ZF) en route to demonstrating their unprovability in PA.

I saw this question just after posting an answer of sorts in the preceding discussion:

Knuth's intuition that Goldbach might be unprovable

As far as I know, the "rapidly growing function" method is not fundamentally different from the "statement allows the theory to construct model of itself" technique. It is an interpretation of the latter method, originally provided by Ketonen and Solovay as a more transparent way of understanding the Paris-Harrington proof (which showed that from the finitary function implied by their variant of Ramsey's theorem, one could construct a model of Peano Arithmetic, and this construction could itself be carried out in PA).

To find a model of PA (or ZF, or other expressive formal system) within a given problem such as Goldbach, it has to support the combinatorics entailed in those formal systems, such as the proof-theoretic ordinals up to $\epsilon_0$ or $\Gamma_0$, or large cardinal combinatorics as in Friedman's work. Theorems of Paris-Harrington, Friedman and others that are unprovable in specific systems refer to structures that are reverse-engineered from the combinatorics appearing in mathematical logic.

Thus, to get PA-unprovability from Goldbach or Riemann conjectures one would have to somehow relate prime number distribution to the extremely rigid combinatorics of either (1) models of PA, (2) the syntax of PA as a formal proof system, or (3) $\epsilon_0$ and the proof-theoretic ordinal analysis of PA. This happens only if things like the Weil conjectures and Riemann hypothesis are the tiniest tip of a structural iceberg in number theory, OR, alternatively, if provable relations to formal systems appear generically in lots of problems all over mathematics, in which case there is a machine for resolving large numbers of open problems (e.g., in ZF) en route to demonstrating their unprovability in PA.

EDIT: regarding the discussion below of rapidly growing functions not seeming to appear in Goldbach, it's not necessary that the/a/any Goldbach function $g(n)$ implicit in the conjecture be fast-growing, only that Peano Arithmetic be able to construct from $g(n)$ something else that is rapid. For example, if you look at the set of $n$ such that $g(n) = 3$, this is a subsequence of something analogous to the twin primes, and conceivably could grow very fast if PA can't prove the Twin Prime Conjecture or its ilk.