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Is there any computable function $f(n)$, which given any integer $n$ has been proven to return either $0$ or $1$ in finite time, and for which the statement "$f(1), f(2), f(3),\ldots$ contains arbitrarily large sequences of $0$'s" has been proved to be undecidable in PA or ZFC?

If not, is there any proof of the existence or non-existence of such a function?

Edit: Is there one which is also morally undecidable?

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What is meant by "morally undecidable"? – r.e.s. Oct 16 '11 at 14:30
up vote 4 down vote accepted

Let $G$ be a Gödel sentence (with intuitive meaning "I cannot be proved"), and take $$f(n)=\cases{0&\text{if }G\text{ has a proof with Gödel number }<n\\1&\text{otherwise}}$$

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Let $G(n)$ be the Goodstein sequence with first element $n$, and take $$f(n)=\cases{0&\text{if }0 \ \text{is an element of} \ G(n) \\1&\text{otherwise.}}$$

EDIT: ZFC proves the statement $\unicode{8220}f(n)=0\text{ for all }n\unicode{8221}$ and also proves that that statement cannot be proved in PA. However, this alone might not imply that PA can't prove the weaker statement $\unicode{8220}f(1),f(2),f(3),\ldots$ contains arbitrarily long sequences of $0\text{s}\unicode{8221}.$ Since I don't know whether PA can in fact prove the latter (weaker) statement, I'm unsure whether my above answer is correct after all.

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Do you mean: undefined otherwise, and we know that doesn't happen, and we were not asked that that prove is in PA? - I don't see how it could work otherwise. – Christian Sievers Jun 21 at 11:48
    
@ChristianSievers - I think you've noticed a flaw in my answer, and I don't see how making $f$ "undefined otherwise" will fix it (as the "otherwise" case seems irrelevant). Although PA can't prove "$f(n)=0$ for all $n$" (as ZFC can), that fact alone does not, of course, imply that PA can't prove the weaker statement "$f(1),f(2),f(3),\ldots$ contains arbitrarily long sequences of $0$s". (I don't know whether PA can prove the latter statement, so I'm not sure what I was thinking when I posted this answer.) – r.e.s. Jun 22 at 20:19
    
Without already using the proof that eventually 0 appears in the Goodstein series, I don't see how to show that your function is computable, while "undefined otherwise" is simple. I thought "arbitrarily long sequences of 0s" implied that all is zero, but that is indeed a different problem. – Christian Sievers Jun 23 at 0:23

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