I am slightly puzzled by the statement: >PA is conservative over HA for pi-0-2 sentneces. I presume this depends on only allowing plus and times as non- logical constants. If one allows Kleene's T-predicate, a counter- example is (forall x)(A(x) or not-A(x)) where A(x) is a sigma-0-1 formula which defines a non-recursive set.