  In this paper we investigate the nature of the singularity of the Ising model
of the 4-dimensional cubic lattice. It is rigorously known that the specific
heat has critical exponent $\alpha=0$ but a non-rigorous field-theory argument
predicts an unbounded specific heat with a logarithmic singularity at $T_c$. We
find that within the given accuracy the canonical ensemble data is consistent
both with a logarithmic singularity and a bounded specific heat, but that the
micro-canonical ensemble lends stronger support to a bounded specific heat. Our
conclusion is that either much larger system sizes are needed for Monte Carlo
studies of this model in four dimensions or the field theory prediction of a
logarithmic singularity is wrong.
