Add an extra symbol A, and the rules that s(A)=42 and 0!=A and forall n: n!=A → s(n) !=A. Then add an exception for A into all the other rules. So s(x)=s(y) → x=y or x=A or y=A.
There are all sorts of ways you could define extra hangers on that didn’t do much in PA or ZFC.
We could describe the laws of physics in this new model. If the result was exactly the same as normal physics from our perspective, ie we can’t tell by experiment, only occamian reasoning favours normal PA.
If I understand it correctly, A is a number which has predicted properties if it manifests somehow, but no rule for when it manifests. That makes it kinda anti-Popperian—it could be proved experimentally, but never refuted.
I can’t say anything smart about this, other than that this kind of thing should be disbelieved by default, otherwise we would have zillions of such things to consider.
Take peano arithmatic.
Add an extra symbol A, and the rules that s(A)=42 and 0!=A and
forall n: n!=A → s(n) !=A. Then add an exception for A into all the other rules. So s(x)=s(y) → x=y or x=A or y=A.
There are all sorts of ways you could define extra hangers on that didn’t do much in PA or ZFC.
We could describe the laws of physics in this new model. If the result was exactly the same as normal physics from our perspective, ie we can’t tell by experiment, only occamian reasoning favours normal PA.
If I understand it correctly, A is a number which has predicted properties if it manifests somehow, but no rule for when it manifests. That makes it kinda anti-Popperian—it could be proved experimentally, but never refuted.
I can’t say anything smart about this, other than that this kind of thing should be disbelieved by default, otherwise we would have zillions of such things to consider.