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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
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up vote 3 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 $$f(n)=\cases{0&\text{if }0 \ \text{is an element of} \ G(n) \\1&\text{otherwise}}$$

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