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https://en.wikipedia.org/wiki/Paris–Harrington_theorem

https://en.wikipedia.org/wiki/Goodstein's_theorem

You can also search for "kirby paris hydra" or the "hydra game" and many interesting results come up. Basically it is a single player game of sorts where the player playing the game always wins "eventually". However, proving that seems to require strength beyond PA.

Based upon my limited knowledge/understanding, there is going to be a fairly large spectrum of strictly increasing (total) recursive functions whose totality will be provable in a stronger system such as ZFC but not in PA. Of course, eventually, there are going to be strictly increasing (total) recursive functions whose totality ZFC won't be able to prove either.

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