Abstract:
The programme of work for this thesis began with the somewhat
genenal intention of parallelling in the context of higher order models
the ultraproduct construction and its consequences as developed in the
literature for first order models. Something of this was, of
course, already available in the ultrapower construction of W.A.J.
Luxemburg used in Non Standand Analysis.
It may have been considered that such a genenal intention was
not likely to yield anything of significance oven and above what was
already available from viewing the higher order situation as a 'many
sorted' first order one and interpreting the first order theory
accordingly. In the event, however, I believe this has proved not to
be so. In particular the substructure concepts developed in Chapter
II of this thesis together with the various embedding theorems and
their applications are not immediately available fnom the first order
theory and seem to be of sufficient worth to warrant developing the
higher order theory in its own terms. This, anyway, is the basic
justification for the approach and content of the thesis.