1. The original example would be Goedel's own example, namely, the statement (formulated in the language of PA) that PA is consistent, i.e., that PA does not pr

1. The original example would be Goedel's own example, namely, the statement (formulated in the language of PA) that PA is consistent, i.e., that PA does not prove 1=0. Less contrived examples are: 2. Goodstein's Theorem https://en.wikipedia.org/wiki/Goodstein%27s_theorem https://en.wikipedia.org/wiki/Goodstein%27s_theorem 3. A theorem stating that every computable strategy is a winning strategy in the hydra game (see https://faculty.baruch.cuny.edu/lkirby/accessible_independence_results.pdf https://faculty.baruch.cuny.edu/lkirby/accessible_independen...) 4. The strengthened finite Ramsey theorem, see https://en.wikipedia.org/wiki/Paris%E2%80%93Harrington_theorem https://en.wikipedia.org/wiki/Paris%E2%80%93Harrington_theor...