[FOM] First-order properties of ordinals

[Ivar Rummelhoff]

Dear everyone, it seems reasonable to believe that the following
result is known:

  Let L be the first-order signature consists of a constant 0 and a
  binary relation symbol <, let n be a natural number, and let alpha
  and beta be ordinals of the form  omega^n · gamma  (defined by
  recursion on gamma).  Then alpha and beta satisfy the same
  Sigma_{2n} L-sentences.

Does anyone on this list happen to have a reference?

(In chapter 2 of my thesis a proof of this - using a variant of
Ehrenfeucht-Fraisse games - is adapted to yield a hierarchy result for
intuitionistic second-order propositional logic.  Comments are
welcome. <http://folk.uio.no/ivarru/research/papers.html#thesis> )

Kind regards,
Ivar Rummelhoff