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If we omit the qualification "natural" from the question, then, of course, the most obvious examples are the arithmetized versions of metamathematical sentences expressing the (absolute) consistency of ZFC, or any axiom-system of set theory far weaker than ZFC within which arithmetic can be developed. Indeed, e.g. Con(ZF) is a sentence of arithmetic that we obtain by Godel-numbering from the metamathematical sentence "there is no proof of 0=1 from ZF." And if ZF is consistent, then Con(ZF) is undecidable in Peano arithmetic with unknown truth value. Actually, on the one hand, Con(ZF) implies Con(PA), and the arithmetical proof of $\lnot$Con(ZF) would yield a direct proof of the inconsistency of ZF.

As far as the "naturalness" condition is concerned, it seems that there will be no easy way to find "natural" sentences of this kind. Indeed, some natural candidates as e.g. the Goldbach conjecture are excluded, since they should be true, if they turn out to be undecidable. More precisely, any $\Pi_1$ sentence $S$ of arithmetic is true whenever $S$ is undecidable. Indeed, if S $S$ is false, then its negation is a true $\Sigma_1$ sentence, and Peano arithmetic proves any true $\Sigma_1$ sentence.

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If we omit the qualification "natural" from the question, then, of course, the most obvious examples are the arithmetized versions of metamathematical sentences expressing the (absolute) consistency of ZFC, or any axiom-system of set theory far weaker than ZFC within which arithmetic can be developed. Indeed, e.g. Con(ZF) is a sentence of arithmetic that we obtain by Godel-numbering from the metamathematical sentence "there is no proof of 0=1 from ZF." And if ZF is consistent, then Con(ZF) is undecidable in Peano arithmetic with unknown truth value. Actually, on the one hand, Con(ZF) implies Con(PA), and the arithmetical proof of not-Con(ZF) $\lnot$Con(ZF) would yield a direct proof of the inconsistency of ZF.

As far as the "naturalness" condition is concerned, it seems that there will be no easy way to find "natural" sentences of this kind. Indeed, some natural candidates as e.g. the Goldbach conjecture are excluded, since they should be true, if they turn out to be undecidable. More precisely, any Pi_1 $\Pi_1$ sentence S $S$ of arithmetic is true whenever S $S$ is undecidable. Indeed, if S is false, then its negation is a true Sigma_1 $\Sigma_1$ sentence, and Peano arithmetic proves any true Sigma_1 $\Sigma_1$ sentence.

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If we omit the qualification "natural" from the question, then, of course, the most obvious examples are the arithmetized versions of metamathematical sentences expressing the (absolute) consistency of ZFC, or any axiom-system of set theory far weaker than ZFC within which arithmetic can be developed. Indeed, e.g. Con(ZF) is a sentence of arithmetic that we obtain by Godel-numbering from the metamathematical sentence "there is no proof of 0=1 from ZF." And if ZF is consistent, then Con(ZF) is undecidable in Peano arithmetic with unknown truth value. Actually, on the one hand, Con(ZF) implies Con(PA), and the arithmetical proof of not-Con(ZF) would yield a direct proof of the inconsistency of ZF.

As far as the "naturalness" condition is concerned, it seems that there will be no easy way to find "natural" sentences of this kind. Indeed, some natural candidates as e.g. the Goldbach conjecture are excluded, since they should be true, if they turn out to be undecidable. More precisely, any Pi_1 sentence S of arithmetic is true whenever S is undecidable. Indeed, if S is false, then its negation is a true Sigma_1 sentence, and Peano arithmetic proves any true Sigma_1 sentence.