Spontaneous symmetry breaking occurs when the action of a theory has a symmetry but the vacuum state violates this symmetry. In that case there will exist a local operator that is non-invariant under the symmetry giving it a nonzero vacuum expectation value. Such non-invariant local operators always have vanishing vacuum expectation values for finite size systems, prohibiting spontaneous symmetry breaking. This occurs because over large timescales, finite systems always transition between all possible ground states, averaging away the expectation value of the operator.