Since it is already bumped, I was wondering if via possible reformulation, the question can be reopened? Many young students like me would benefit immensely; although this is a research level site, the fact is to explain the concepts in expository terms, there is no other better forum than this one with experts.
I am willing to change the question to: Examples of applications of foundational theorems
(1 each, CW) with OP reading:
Formal definition of Goodstein's theorem: "Every Goodstein sequence eventually terminates at 0. The Goodstein sequence G(m) of a number m is a sequence of natural numbers. The first element in the sequence G(m) is m itself. To get the next element, write m in hereditary base 2 notation, change all the 2s to 3s, and then subtract 1 from the result; this is the second element of G(m). To get the third element of G(m), write the second element in hereditary base 3 notation, change all 3s to 4s, and subtract 1 again. Continue until the result is zero, at which point the sequence terminates."
Example by analogy: "Laurie Kirby and Jeff Paris gave an interpretation of the Goodstein's theorem as a hydra game: the "Hydra" is a rooted tree, and a move consists of cutting off one of its "heads" (a branch of the tree), to which the hydra responds by growing a finite number of new heads according to certain rules. The Kirbyâ€“Paris interpretation of the theorem says that the Hydra will eventually be killed, regardless of the strategy that Hercules uses to chop off its heads, though this may take a very, very long time."