Thank you for this answer. You also helped make another point of confusion clearer: When I read the op I wasn’t sure what the interpretation of number theory is. There must be one or the incompleteness theorem wouldn’t apply but what is it?
Based on your comment here I understand this better and it makes more sense why people might disagree about for example the continuum hypothesis being “reasonable”. While mostly everyone agrees on the standard ZFC axioms not everyone agrees on the interpretation of ZFC.
Then there must also be hardliners who take ZFC as the truth without interpretation and thus do not allow statements independent of it as they are not provable. It would be interesting to see what proofs they miss out on. (Not requesting anyone to list them for me, just thinking out loud.)
Thank you for this answer. You also helped make another point of confusion clearer: When I read the op I wasn’t sure what the interpretation of number theory is. There must be one or the incompleteness theorem wouldn’t apply but what is it?
Based on your comment here I understand this better and it makes more sense why people might disagree about for example the continuum hypothesis being “reasonable”. While mostly everyone agrees on the standard ZFC axioms not everyone agrees on the interpretation of ZFC.
Then there must also be hardliners who take ZFC as the truth without interpretation and thus do not allow statements independent of it as they are not provable. It would be interesting to see what proofs they miss out on. (Not requesting anyone to list them for me, just thinking out loud.)