Following this question I still do not get clearly the difference between defining exponentiation in PA but impossiblity of recursively define multiplication in Presburger Arithmetics
I was thinking one can do things like define base case Multi(x, y, z) where y is 0 and z is 0, and then define Multi(x,y,z) being there exist z which is the sum of z2 from Multi(x, y2, z2) and y, where y2 is the predecessor of y?
Apparently for some reason this is not allowed? I assume this is to do with: "Presburger arithmetic is not strong enough to quantify over sequences", but why? what is the reason? is it because the induction schema is different between the two arithmetics?