Every substance is necessarily infinite
Formal Statement
Every substance is necessarily infinite. If a substance were finite, it would be limited by another substance of the same nature (Def.2). But that other substance would also have to exist necessarily (P7), and then two substances would share an attribute — contradicting P5. Therefore no substance can be finite; every substance is infinite.
In Plain Language
Finite means limited by something of the same kind — a finite body is bounded by a larger body, a finite thought by a larger thought. But if a substance were limited by another substance of the same attribute, you would have two substances with the same attribute, which P5 forbids. And since substance must exist (P7), the only option left is that it exists as infinite. Finitude is for modes; substance is necessarily unlimited.
Why This Follows
P7 (gs-09) established that existence belongs to substance's nature. P5 (gs-07) established that no two substances share an attribute. If a substance were finite, Def.2 says it would be limited by another of the same kind — but that other substance would also necessarily exist (P7), violating P5. So substance must be infinite.
Every substance is necessarily infinite — finite substance is impossible.
Connected Concepts
Spinoza defines "finite" as "limited by another of the same nature." Is that the only meaningful sense of finitude, or could something be finite in another way?