  Topoi are categories which have enough structure to interpret higher order
logic. They admit two notions of morphism: logical morphisms which preserve all
of the structure and therefore the interpretation of higher order logic, and
geometric morphisms which only preserve only some of the structure and
therefore only some of the interpretation of higher order logic. The question
then arises: what kinds of higher order theories are preserved by geometric
morphisms? It is known that certain first order theories called internal
geometric theories are preserved by geometric morphisms, and these admit what
are known as classifying topoi. Briefly, a classifying topos for an internal
geometric theory $\mathbb{T}$ in a topos $\mathcal{E}$ is a topos
$\mathcal{E}[\mathbb{T}]$ such that models of $\mathbb{T}$ in any topos
$\mathcal{F}$ with a geometric morphism to $\mathcal{E}$ are in one to one
correspondence with geometric morphisms from $\mathcal{F}$ to
$\mathcal{E}[\mathbb{T}]$ over $\mathcal{E}$. One useful technique for showing
that a higher order theory $\mathcal{T}$ is preserved by geometric morphisms is
to define an internal geometric theory $\mathbb{T}$ of "bad sets" for
$\mathcal{T}$ and show that $\mathcal{T}$ is equivalent to the higher order
theory which says "the classifying topos for $\mathbb{T}$ is degenerate". We
set up a deduction calculus for internal geometric theories and show that it
proves a contradiction if and only if the classifying topos of that theory is
degenerate. We use this result to study a variant of the higher order theory of
Dedekind finite objects and the higher order theory of field objects considered
as ring objects with no non-trivial ideals.
