- Our understanding of God is a being than which no greater can be conceived.
- The idea of God exists in the mind.
- A being which exists both in the mind and in reality is greater than a being that exists only in the mind.
- If God only exists in the mind, then we can conceive of a greater being—that which exists in reality.
- A being of whom more exist in reality is greater than a being of whom fewer exist in reality.
- If God is a being of Whom n exist, then we can conceive of a greater being—a being of Whom n + 1 exist.
- We cannot be imagining something that is greater than God.
- Therefore, by induction an infinite number of Gods exist.
I maintain that this version is entirely acceptable--inevitable, even--by the logic of the original argument, but even if you disagree, it makes the fallacy of the ontological argument clearer. There is no intrinsic difference between something of which n exist and something of which n + 1 exist--it does not change the nature of the thing and certainly does not make it greater. Notice how awkwardly I had to word it--"a being of Whom n exist"--to make quantity even sound like a property of something. If n = 0, this means that the existence or nonexistence of something does not affect its nature. In other words, "existence" is not a property you can apply to something like you can attributes of greatness--God's power, knowledge, love, etc.
An analogy from my native field of computer science. It is common to define classes, which can then be instantiated into individual objects. It's similar to the idea of Platonic forms--I could define a "Circle" class, analogous to the abstract conception of a circle, then then from that class make some Circle objects with definite radii, positions, etc. Each Circle object has its own particular properties, but the Circle class (abstract circle) also has some intrinsic properties--the formulas for perimeter, area, and so on. It is part of the nature of a circle that its perimeter is 2π times its radius. It is not part of the nature of a circle--not in anything like the same way--that any particular circle or circular object exists. The existence or nonexistence of real circles does nothing to change the nature of the abstract circle.
And so with God. Anselm is taking an extrinsic property of God--existence or nonexistence--and trying to reason about it as if it were an intrinsic property--like God's justice. Intrinsic properties are true of something even when reasoning about it in the abstract and if it doesn't exist, extrinsic properties like existence or position only apply to things that already exist. Generalizing existence (0 or 1?) into numerical quantity (how many?) makes this fallacy much more obvious. If God is better if He exists rather than not existing, then wouldn't God be greater still if there were two of Him? And so on, by induction, until you have proved the existence of an infinite number of Gods.