  In this paper we show that the incompressible Hall-MHD system without
resistivity is not globally in time well-posed in any Sobolev space
$H^{m}(\mathbb{R}^3)$ for any $m>\frac{7}{2}$. Namely, either the system is
locally ill-posed in $H^{m}(\mathbb{R}^3)$, or it is locally well-posed, but
there exists an initial data in $H^{m}(\mathbb{R}^3)$, for which the
$H^{m}(\mathbb{R}^3)$ norm of solution blows-up in finite time if $m>7/2$.
  In the latter case we choose an axisymmetric initial data
$u_0(x)=u_{0r}(r,z)e_r+ b_{0z}(r,z)e_z$ and
$B_0(x)=b_{0\theta}(r,z)e_{\theta}$, and reduce the system to the axisymmetric
setting. If the convection term survives sufficiently long time, then the Hall
term generates the singularity on the axis of symmetry and we have
  $ \lim\sup_{t\to t_*} \sup_{z\in \Bbb R} |\partial_z\partial_r
b_\theta(r=0,z)|=\infty$ for some $t_*>0$.
