  We show that the first order theory of the homeomorphism group of a compact
manifold interprets the full second order theory of countable groups of
homeomorphisms of the manifold. The interpretation is uniform across manifolds
of bounded dimension. As a consequence, many classical problems in group theory
and geometry (e.g.~the linearity of mapping classes of compact $2$--manifolds)
are encoded as elementary properties homeomorphism groups of manifolds.
Furthermore, the homeomorphism group interprets the Borel and projective
hierarchies of the homeomorphism group. Finally, we show that the collection of
sentences that isolate the homeomorphism group of a particular manifold, or
that isolate the homeomorphism groups of manifolds in general, is not definable
in arithmetic, and that membership of particular sentences in these collections
cannot be proved in ZFC.
