A section on the Ising model in the text "Introduction to modern statistical mechanics" by Chandler states the following:
"We consider a system of $N$ spins arranged on a lattice. In the presence of a magnetic field, $H$, the energy of the system in a particular state, $\nu$, is $$E_{\nu} = \sum_{i=1}^{N}H \mu s_i + (\text{energy due to interactions between spins}), $$ where $s_i = \pm 1$. A simple model for the interaction energy is $$-J\sum_{ij} ' s_i s_j$$ where $J$ is called a coupling constant, and the primed sum extends over nearest-neighbor pairs of spins. The spin system with this interaction energy is called the Ising Model.
Notice that when $J > 0$, it is energetically favorable for neighboring spins to be aligned. Hence, we might anticipate that for low enough temperature, this stabilization will lead to a phenomenon called spontaneous magnetization. "
Questions:
In the last paragraph, by "energetically favorable if $J > 0$", is the implication that, assuming $J > 0$ and aligned neighbouring spins, it follows that $-J\sum_{ij}'s_i s_j$ would be negative and hence the interaction energy reduces the total energy $E_{\nu}$ thus moving in the direction of minimum energy, which a stable system in equilibrium tends to after being perturbed. Is this the correct reasoning?
Lastly, why would we anticipate that for low enough temperature, this stabilization will lead to spontaneous magnetization?