For simplicity, say there were only two unmoved movers, β & ψ. They would each be an actus purus, by definition. They would both likewise be necessary and eternal.
Neither of them could influence the other, obviously. So, they couldn’t do or know anything about each other, and would not therefore be either omnipotent or omniscient. Nor could either one of them be properly understood as ultimate, because by the definition of ‘ultimate,’ there can be only one ultimate. So neither of them could be God (that’s why I didn’t label them α & ω).
In order both to exist, β & ψ need to be somehow different from each other. There must be at least one property that they do not both equally share. This is Leibniz’ Principle of the Identity of Indiscernibles: if the referent of “A” and the referent of “B” have exactly the same properties, then both “A” and “B” refer to the same entity (e.g., “3” and “4 – 1” refer to the same quantity). Put another way, you can’t have more than one being that exists in exactly the same way.
Simplifying again, say that β & ψ each have only two properties: the property of necessity, and either p or ~p. They are both necessary, but β is p, while ψ is ~p. Here’s the question: is it possible for β to be ~p?
Yes: if a necessary being like ψ can be ~p, then it is possible for an otherwise identical necessary being such as β to be ~p – after all, ψ has managed it – and β could therefore possibly be ~p. But this means that β is not necessarily p. And this means that β is not entirely necessary, but rather partially contingent. But a partially contingent being *just is* a contingent being.
The same goes for ψ: if β can be p, so can ψ; so ψ is only contingently ~p, and is therefore not a necessary being, but rather a contingent being.
Contingencies require causes other than themselves to be just the things that they are (when, of course, they might possibly have been otherwise). So, they cannot be unmoved movers.
Thus there can be at most one unmoved mover.