  Two general unsolvability arguments for interacting bosonic quantum field
theories are presented, based on Dyson-Schwinger equations on the lattice and
cardinality considerations. The first argument is related to the fact that, on
a lattice of size N, the system of lattice Dyson-Schwinger equations closes on
a basis of "primitive correlators" which is finite, but grows exponentially
with N. By properly defining the continuum limit, one finds for N -> infinity a
countably-infinite basis of the primitive correlators. The second argument is
that any conceivable exact analytic calculation of the primitive correlators
involves, in the continuum limit, a linear system of coupled partial
differential equations on an infinite number of unknown functions, namely the
primitive correlators, evolving with respect to an infinite number of
independent variables.
