Working in HST

It follows from Theorem 1.4.6(ii) that, under certain conditions, sets of *-ordinals, with all proper initial segments internal, are necessarily internal themselves. However if internality is replaced by the stronger condition of h-standardness, where h is a hyperfinite number in *N, then the result is not true any more. As an example of how advanced model theoretic arguments can be naturally dealt with in HST we present here two counterexamples of this kind, one originally from a known recent paper by Andreas Blass and another (simpler) one also communicated by Andreas:
Blass (pdf)
Blass (ps)