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Peano arithmetic is enough, because Peano arithmetic encodes computation
This is one of a pair of questions trying to understand this comment on the xkcd forum contest My number is bigger than yours!. For a definition of Goodstein sequences, see this question. Let $G(n)...
I will show this by demonstrating that PA can construct a proof of length $O(\log^*(n) \log(\log^*(n)))$, where the funny function is the very slowly growing iterated logarithm. Now go back and look at the Goodstein Sequence, and convince yourself that hereditary base notation represents ordinals written out in Cantor normal form. The impossibility of proving such statements true or false is what gives us things like Gödel incompleteness, the unsolvability of the Halting problem, and so on.
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