St. Anselm's Ontological Argument

For this week’s faith and philosophy discussion, I thought we could look at one of the most enduring arguments in philosophy: how do we prove God exists? This is known as the ontological argument, and has been the focus of several great arguments, including the Watchmaker Analogy (see page 40 of our eBook, it is a great read!).
St. Anselm formulated the first ontological argument that God exists in reality because the concept of God does not bear internal contradictions. Here are the major points to his argument:
God exists in our understanding. This means that the concept of God resides as an idea in our minds.
God is a possible being, and might exist in reality. He is possible because the concept of God does not bear internal contradictions.
If something exists exclusively in our understanding and might have existed in reality then it might have been greater. This simply means that something that exists in reality is perfect (or great). Something that is only a concept in our minds could be greater by actually existing.
Suppose (theoretically) that God only exists in our understanding and not in reality.
If this were true, then it would be possible for God to be greater than he is (follows from premise #3).
This would mean that God is a being in which a greater is possible.
This is absurd because God, a being in which none greater is possible, is a being in which a greater is possible. Herein lies the contradiction.
Thus it follows that it is false for God to only exist in our understanding.
Hence God exists in reality as well as our understanding (“Philosophical Proofs on the Existence of God”, 2015, para 4).
I would love to hear your thoughts on St. Anselm's ontological argument.
Sincerely,
Elizabeth
Reference:
“Philosophical Proofs on the Existence of God” (2015). http://www(dot)saintaquinas(dot)com/philosophy.html