Is there any natural* statement S about the natural integers such that if PA contains no contradictions then neither PA+S nor PA+not S contains a contradiction?

If unknown, where can I read about the philosophical views on it?

*By natural I mean not a logical trick or non-constructive existence proof such as Rosser's sentence, but a clear statement such as the collatz conjecture

Is there any natural* statement S about the natural integers such that if PA contains no contradictions then neither PA+S nor PA+not S contains a contradiction?

If unknown, where can I read about the philosophical views on it?

*By natural I mean not a logical trick or non-constructive existence proof such as Rosser's sentence, but a clear statement such as the Collatz conjecture.

Is there any statement S about the natural integers such that if PA contains no contradictions then neither PA+S nor PA+not S contains a contradiction?

If unknown, where can I read about the philosophical views on it?

Is there any natural* statement S about the natural integers such that if PA contains no contradictions then neither PA+S nor PA+not S contains a contradiction?

If unknown, where can I read about the philosophical views on it?

*By natural I mean not a logical trick or non-constructive existence proof such as Rosser's sentence, but a clear statement such as the collatz conjecture