  This paper investigates the absolute values on $\mathbb{Z}$ valued in the
upper reals (i.e. reals for which only a right Dedekind section is given).
These necessarily include multiplicative seminorms corresponding to the finite
prime fields $\mathbb{F}_p$. As an Ostrowski-type Theorem, the space of such
absolute values is homeomorphic to a space of prime ideals (with co-Zariski
topology) suitably paired with upper reals in the range $[-\infty, 1]$, and
from this is recovered the standard Ostrowski's Theorem for absolute values on
$\mathbb{Q}$.
  Our approach is fully constructive, using, in the topos-theoretic sense,
geometric reasoning with point-free spaces, and that calls for a careful
distinction between Dedekinds vs. upper reals. This forces attention on
topological subtleties that are obscured in the classical treatment. In
particular, the admission of multiplicative seminorms points to connections
with Berkovich and adic spectra. The results are also intended to contribute to
characterising a (point-free) space of places of $\mathbb{Q}$.
