Phase transitions of interacting particles in statistical mechanics can be seen in computer simulations, but cannot be mathematically proven. A few 'spin' models, chiefly the 2-dimensional Ising model, can be proven to show a transition. We show the underlying difficulty, and discuss a new class of models in which some examples can be proven to show transitions. Relevance to scissors congruence and the Banach Tarski paradox will be discussed if there is time.