Anselm's Ontological Argument for God's Existence As a Reductio ad Absurdum Argument

Today I begin teaching the first of a number of philosophical arguments for God's existence in my MCCC Philosophy of Religion classes. I'll make posts on these to assist students in understanding. Remember: 1) Understand; 2) Evaluate. (As opposed to Facebook discussions which almost always go like this: 1) Evaluate. With no understanding.  :(

Anselm's famous Ontological Argument for God's Existence can be formulated this way:

1. I have an idea of a being a greater than which can not be conceived.
2. Therefore, God exists.

Anselm's argument is a reductio ad absurdum form of argument. In a reductio argument the opponent's view is accepted and then shown to be absurd. We can then formulate Anslem's OA this way:

1. Persons have the idea of a greatest possible being.
2. Suppose the greatest exists only as an idea in the mind.
3. Existence in reality is greater than existence only in the mind.
4. Therefore, we can conceive of a being greater than the greatest possible being, that is, a being that also exists in reality.
5. But there can be no being than the greatest possible being.
6. Therefore, since 4 and 5 together yield a self-contradiction, the greatest possible being exists in reality.

(From Reason and Religious Belief: An Introduction to the Philosophy of Religion, Fourth Edition, p. 93)

To my former Philosophy of Religion students - you should understand this, right?

There seems to be something quite wrong with this argument. "Yet, despite these suspicions, philosophers have found it difficult to discern precisely where the argument is vulnerable and to make a strong case that this point of suspicion is a vice rather than a virtue." (Op. cit., 93)