I am shocked after reading this: http://googology.wikia.com/wiki/Finite_promise_games(adsbygoogle = window.adsbygoogle || []).push({});

So, lets take strong Goodstein function. It is total, but this fact is unprovable in Peano Arithmetics. No problem, I just understand that PA is too weak. Goodstein function is total, just take stronger theory to prove it.

Now, set theory, ZF(C) or NBG is so strong that nobody dares to prove it's consistency, because it would require even stronger theory. There are many flavors of set theory, take CH/GCH (or negation) and combine with any large cadinal axiom (or negation of it), it will be consistent. For me it was like a game, because, I had assumed, there are no real consequences.

However, after reading the article I see that there are some finite promise games (sequences), which terminate only if we assume the existence of Mahlo cardinal. I see striking resemblance to Goodstein sequences. So I have to admit that such sequences do terminate (because no contre-example could be made, otherwise it would be decidable without additional axioms), but ZFC is not powerful enough to prove it. So, at least in some sense, Mahlo is true, it is not just the what-if game...

I am deeply shocked...

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# I Finite promise games, shock of my life

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