We know that Dedekind in 1888 proved that second order peano arithmetic(PA2) is categorical. My question is, why is it important? Does it have any mathematical, philosophical or foundamental consequences? i.e if PA2 wasn't categorical, what would we lose?!
I think one of the result of it is that second order theoris don't need to be complete; because: By Godel's incompleteness, Let G be Godel's sentence which is true in N but is not provable by PA2. Since N is the only model for PA2, PA2 satisfies G. But PA2 doesn't prove G.
But the question is,are Existence of categorical theoris essential for showing that second order theories don't need to be complete?
Isn't there any other way to show that?
If yes, two questions arises:
- What is it?
- Then what is the importance of categoricity?
If No, is there any justification?!