  The spectral principle of Connes and Chamseddine is used as a starting point
to define a discrete model for Euclidean quantum gravity. Instead of summing
over ordinary geometries, we consider the sum over generalized geometries where
topology, metric and dimension can fluctuate. The model describes the geometry
of spaces with a countable number $n$ of points, and is related to the Gaussian
unitary ensemble of Hermitian matrices. We show that this simple model has two
phases. The expectation value $<n>$, the average number of points in the
universe, is finite in one phase and diverges in the other. We compute the
critical point as well as the critical exponent of $<n>$. Moreover, the
space-time dimension $\delta$ is a dynamical observable in our model, and plays
the role of an order parameter. The computation of $<\delta>$ is discussed and
an upper bound is found, $<\delta> < 2$.
