I’m still not sure what you want to say. It’s a necessary property of natural numbers that they can be reached from iterating the successor function. That condition can’t be expressed in first-order logic, so it can’t be proved and it holds in some models and in others it doesn’t. It’s like trying to define “cat” by stating that it’s an animal. This is not a sufficient definition.
You’re the one who brought up the natural numbers, I’m just saying they’re not relevant to the discussion because they don’t satisfy the uniqueness thing that OP was talking about.
I’m still not sure what you want to say. It’s a necessary property of natural numbers that they can be reached from iterating the successor function. That condition can’t be expressed in first-order logic, so it can’t be proved and it holds in some models and in others it doesn’t. It’s like trying to define “cat” by stating that it’s an animal. This is not a sufficient definition.
You’re the one who brought up the natural numbers, I’m just saying they’re not relevant to the discussion because they don’t satisfy the uniqueness thing that OP was talking about.