In the presence of primordial magnetic fields the oscillating cosmic a
xion field drives an oscillating electric field. The ensuing dissipati
on of axions is found to be inversely proportional to the conductivity
of the primordial plasma. This counterintuitive result is essentially
equivalent to ''Zeno's paradox'' or the ''watched-pot effect'' of qua
ntum mechanics. It implies that the standard predictions of the cosmic
axion density remain unaltered even if primordial magnetic fields are
strong.