My understanding is you can specifically and formally define the natural numbers with addition and multiplication, although multiplication means the language is no longer decidable.
You can define natural numbers with just addition ( Presburger arithmetic ) and it’s decidable.
Im not sure how undecidable <=> “will define things that are similar to natural numbers but are not” but maybe I am missing something
If a sentence S is undecidable from your axioms for the natural numbers then there are two models A and B satisfying those axioms where A satisfies S and B satisfies not S. So which one is the standard natural numbers, is it A or B?
Either A or B will be an example of something that satisfies your definition of natural numbers and yet is not the natural numbers.
My understanding is you can specifically and formally define the natural numbers with addition and multiplication, although multiplication means the language is no longer decidable.
You can define natural numbers with just addition ( Presburger arithmetic ) and it’s decidable.
Im not sure how undecidable <=> “will define things that are similar to natural numbers but are not” but maybe I am missing something