  Using the lattice-theoretic version of the Euler characteristic introduced by
V. Klee and G.-C. Rota in the Sixties, we define the Euler characteristic of a
formula in G\"{o}del logic (over finitely or infinitely many truth-values). We
then prove that the information encoded by the Euler characteristic is
classical, i.e. coincides with the analogous notion defined over Boolean logic.
Building on this, we define many-valued versions of the Euler characteristic of
a formula $\varphi$, and prove that they indeed provide information about the
logical status of $\varphi$ in G\"{o}del logic. Specifically, our first main
result shows that the many-valued Euler characteristics are invariants that
separate many-valued tautologies from non-tautologies. Further, we offer an
initial investigation of the linear structure of these generalised
characteristics. Our second main result is that the collection of many-valued
characteristics forms a linearly independent set in the real vector space of
all valuations of G\"{o}del logic over finitely many propositional variables.
