  A finite element elasticity complex on tetrahedral meshes is devised. The
$H^1$ conforming finite element is the smooth finite element developed by
Neilan for the velocity field in a discrete Stokes complex. The symmetric
div-conforming finite element is the Hu-Zhang element for stress tensors. The
construction of an $H(\textrm{inc})$-conforming finite element for symmetric
tensors is the main focus of this paper. The key tools of the construction are
the decomposition of polynomial tensor spaces and the characterization of the
trace of the $\textrm{inc}$ operator. The polynomial elasticity complex and
Koszul elasticity complex are created to derive the decomposition of polynomial
tensor spaces. The trace of the $\textrm{inc}$ operator is induced from a
Green's identity. Trace complexes and bubble complexes are also derived to
facilitate the construction. Our construction appears to be the first
$H(\textrm{inc})$-conforming finite elements on tetrahedral meshes without
further splits.
