Question

I read your explanation of the ontological argument and didn’t completely understand it.
If I understood correctly, it is made up of two parts (chapters 2 and 3 of the Proslogion):
A. A thing that exists is greater than one that does not exist. One can conceive of God as existing. Therefore God must be defined as existing, since otherwise one could conceive of something greater than Him.
B. A thing that exists necessarily is greater than one that exists but not necessarily. One can conceive of God as existing necessarily. Therefore God must be defined as existing necessarily, since otherwise one could conceive of something greater than Him.
As for claim A, I didn’t understand why it proves that God exists in this world of ours. It only says that only a God who exists is God, since only He is the greatest thing that can be conceived. It doesn’t say that only a God who exists in our world is God. (It seems to me that you yourself addressed this claim, and indeed explained that the main proof is claim B.)
Claim B indeed does prove that He exists in our world, since His existence is necessary, but I didn’t understand why the assumption is correct that we can conceive of God as something that exists necessarily. Necessary existence is not an attribute that we simply decide to grant to whomever we want. It must stem specifically from logical necessity. In order to apply that assumption to God, we need to find such a necessity—but if we bring a proof like that, it would itself be the proof of God’s existence, without the ontological argument.
 
In any case, thanks in advance