  We show that in an ultraproduct of finite fields, the mod-$n$ nonstandard
size of definable sets varies definably in families. Moreover, if $K$ is any
pseudofinite field, then one can assign "nonstandard sizes mod $n$" to
definable sets in $K$. As $n$ varies, these nonstandard sizes assemble into a
definable strong Euler characteristic on $K$, taking values in the profinite
completion $\hat{\mathbb{Z}}$ of the integers. The strong Euler characteristic
is not canonical, but depends on the choice of a nonstandard Frobenius. When
$\operatorname{Abs}(K)$ is finite, the Euler characteristic has some funny
properties for two choices of the nonstandard Frobenius.
  Additionally, we show that the theory of finite fields remains decidable when
first-order logic is expanded with parity quantifiers. However, the proof
depends on a computational algebraic geometry statement whose proof is deferred
to a later paper.
