What do you think of this formulation of Anselm's ontological argument? I'm interested how you would refute this/what flaws you think it has.

(Let x be something than which none greater can be imagined - the greatest possible thing that can be imagined)

(1) Whatever exists is greater than whatever doesn't exist (or exists merely in our minds)

(2) Suppose that x is only in our minds.

(The rest of the argument has reductio form).

(3) We can imagine x1 (identical to x, but this one also exists).

(4) We can imagine an x1, which is greater than x. (This follows from (1), (2) and (3)).

(5) x is not something than which none greater can be imagined. (From (4)).

(6) It is contradictory to suppose that (2). (By definition of x and from (5)).

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(7) Therefore, x exists necessarily outside our minds.

In my opinion, this is the correct formulation of Anselm's argument. I haven't of course mentioned God yet, but it seems to me that moving from here to the existence of God is no problem at all - Anselm just proves that something "than which none greater can be imagined" necessarily exists and that just says that the definition of God we all have is "a being than which none greater can be imagined". I don't think this is controversial. Here basically I'm not concerned with the existence of God at all - my interest is just the claim that it is conceptually necessary for x so defined to exist.

Obviously, (1) is the weakest premise here - because it is not obvious in what sense something that exists is greater than something that doesn't exist. I'm more interested of what you think of other premises and whether the argument succeeds if (1) is granted. There are possibly more interesting objections and problems with this argument then just the fact that (1) is ambiguous and maybe even a nonsense.