  We study the Sierpinski object $\Sigma$ in the realizability topos based on
Scott's graph model of the $\lambda$-calculus. Our starting observation is that
the object of realizers in this topos is the exponential $\Sigma ^N$, where $N$
is the natural numbers object. We define order-discrete objects by
orthogonality to $\Sigma$. We show that the order-discrete objects form a
reflective subcategory of the topos, and that many fundamental objects in
higher-type arithmetic are order-discrete. Building on work by Lietz, we give
some new results regarding the internal logic of the topos. Then we consider
$\Sigma$ as a dominance; we explicitly construct the lift functor and
characterize $\Sigma$-subobjects. Contrary to our expectations the dominance
$\Sigma$ is not closed under unions. In the last section we build a model for
homotopy theory, where the order-discrete objects are exactly those objects
which only have constant paths.
