  Observations of the cores of nearby galaxy clusters show H$\alpha$ and
molecular emission line filaments. We argue that these are the result of {\em
local} thermal instability in a {\em globally} stable galaxy cluster core. We
present local, high resolution, two-dimensional magnetohydrodynamic simulations
of thermal instability for conditions appropriate to the intracluster medium
(ICM); the simulations include thermal conduction along magnetic field lines
and adiabatic cosmic rays. Thermal conduction suppresses thermal instability
along magnetic field lines on scales smaller than the Field length ($\gtrsim$10
kpc for the hot, diffuse ICM). We show that the Field length in the cold medium
must be resolved both along and perpendicular to the magnetic field in order to
obtain numerically converged results. Because of negligible conduction
perpendicular to the magnetic field, thermal instability leads to fine scale
structure in the perpendicular direction. Filaments of cold gas along magnetic
field lines are thus a natural consequence of thermal instability with
anisotropic thermal conduction. Nonlinearly, filaments of cold ($\sim 10^4$ K)
gas should have lengths (along the magnetic field) comparable to the Field
length in the cold medium $\sim 10^{-4}$ pc! Observations show, however, that
the atomic filaments in clusters are far more extended, $\sim 10$ kpc. Cosmic
ray pressure support (or a small scale turbulent magnetic pressure) may resolve
this discrepancy: even a small cosmic ray pressure in the diffuse ICM, $\sim
10^{-4}$ of the thermal pressure, can be adiabatically compressed to provide
significant pressure support in cold filaments. This is qualitatively
consistent with the large population of cosmic rays invoked to explain the
atomic and molecular line ratios observed in filaments.
