Original Position has kindly accepted the task of defending the following argument.

1. God exists in the understanding but not in reality. (Assumption for reductio)
   
2. Existence in reality is greater than existence in the understanding alone. (Premise)
   
3. A being having all of God's properties plus existence in reality can be conceived. (Premise)
   
4. A being having all of God's properties plus existence in reality is greater than God. (From (1) and (2).)
   
5. A being greater than God can be conceived. (From (3) and (4).)
   
6. It is false that a being greater than God can be conceived. (From definition of “God”.)
   
7. Hence, it is false that God exists in the understanding but not in reality. (From (1), (5), (6).)
   
8. God exists in the understanding. (Premise, to which even the Fool agrees.)
   
9. Hence God exists in reality. (From (7), (8).)

Noncognitivists might disagree.

Yeah, well, they're wrong.

EDIT: More substantively, the argument obviously fails if this premise is false, but it is accepted by most atheists, right? So at minimum this argument would show that those atheists have inconsistent beliefs.

go on...

edit: I am not convinced that (at least some of) the descriptions of god are meaningful or well defined concepts. So it might be a minority view among atheists, but I am not sure it is obvious that we CAN conceive of a being of which no greater being is possible. Yes it is a grammatically correct sentence, but is it necessarily more than that? At the very least, I would need some pretty indepth discussion of "being", "greater", "conceive" and the like to know what is actually being talked about. Perhaps that is just philosophical immaturity on my part though...

Here is my conception of god: the being than which nothing greater can be conceived. If you wish to show how this conception is incoherent, feel free. Otherwise, I don't see why I have to prove that it is coherent. I understand all the terms involved and there are no obvious contradictions. What's the problem?

This is unconvincing. How is existence in reality "greater" than existence in the understanding alone? Who is to say that the greatest measure of existance is to exist in reality? I could probably think of a few things that would be greater about existing in understanding than in reality (I won't attempt to lest they undermine the point that SOMEONE could probably come up with reasons why the property of existing in understanding is superior to existing in reality).

It seems to me that the only way this argument works is if we define the greatest type of existence as that which exists outside of the understanding, but why would we accept it?

You use the word "greater". Okay, well I have a loose intuitive sense of what that word means, but I don't see how that meaningfully applies to the topic at hand. You are essentially asserting the existence of a partial ordering on the set of all "beings".

For instance, is an elephant greater than a dog? I can say this sentence in english and it appears to make gramatical sense, but I don't have any knowledge of some agreed upon ordering that lets me compare and say one is greater than the other. We could define our ordering to be, say, size in which case I could say which the elephant is bigger than the dog, but I don't know what sense it would be greater. Of course, a partial ordering is not the same as a complete ordering so perhaps you can compare the "god" being (whatever that is) to every other one, but not compare two arbitrary beings (like an elephant and a dog). But hopefully this demonstrates how it is not at all clear what is meant by "greater" or why we can necessarily conceive of this partial ordering on the set of all beings or this god being or anything else.

There is some bait and switch going on in premise 3. "God" is defined as "The greatest conceivable being" [I missed adding this to the OP, maybe OrP can add it?] If I grant premise 2 then it is incoherent to say that I can accurately conceive of the greatest conceivable being in understanding without presupposing the property of existence. Any being of which I can conceive in understanding that does not exist is great, sure, but not the greatest.

[edit: removed some stuff while I think on it a bit longer]

I would say the relevant sense is that some things are greater as beings than others. We could understand this in terms of skills, abilities, or accomplishments. For instance, I would say that Albert Einstein is a greater man than I because he was more skilled, had greater natural ability, and accomplished more than I have.

We could understand this in terms of value or importance. For instance, I would say that Albert Einstein was a greater man than Babe Ruth because what he did and his excellences as a person were more important and/or valuable than the accomplishments and excellences of Babe Ruth.

Applying this to God, I would say that this definition would then mean that God has all of those excellences or valuable qualities to a maximal degree possible consistent with having other excellences. So we typically think of knowledge and wisdom as being valuable or important qualities--so God would have the maximal amount of those qualities. And so on.

Three comments. First, your inability to decide whether an elephant is greater than a dog doesn't necessarily come from your not knowing how to order the greatness of beings. It might just be a result of elephants and dogs have a closely similar amount of greatness. We can see this by looking at other cases--say humans and typewriters. I think most people would feel fairly comfortable saying that humans are greater than typewriters.

Second, your claim is that my conception of god is incoherent. This means that you have to show not only that you don't know how to order the set of beings, but that it cannot be ordered. You haven't even attempted to show this.

Third, I think we are getting a bit far afield from the ontological argument. The argument only becomes relevant if we accept the coherence of the concept of god--otherwise it is sort of irrelevant. It is sort of like me saying that the ontological argument fails because the god would allow evil and evil is allowed....

You don't think it is a characteristic of beings that existing in reality rather than as an idea makes them greater? For instance, isn't an actual horse greater than the idea of a horse? Isn't an actual $100 greater than the idea of $100? Isn't actually winning the WSOP greater than the idea of winning the WSOP?

Is an actual serial-killer greater than the idea of a serial-killer?

Nah, your argument is obviously unsound in a way that mine is not. For instance, we obviously cannot conceive of a being that has all of Allah's properties plus existence in reality since one of Allah's properties is that she doesn't exist in reality. Also, presumably you agree with the conclusion in your argument. Yahweh would exist in reality. Might not be god--could be Barack Obama for all we know--but presumably there is something that exists that is the greatest thing we can can conceive of that actually exists.

More generally, if you want to show that my argument fails, you'll have to do so by showing that the ambiguity exists in my definition of god, not yours.

In some ways, yes, although in others no. Overall, I'm not sure. However, there are obviously going to be some paradoxes that pop up here (is the idea of non-existence greater than actual non-existence?). Does that mean we should give up on this idea? Are we really going to say that a tennis ball is just as valuable or great or important as a human being?

Replace <purple and yellow striped brontosaurs> with anythig as required.

Nope. In your argument, premise 6 is clearly false, whereas in my argument, it is clearly true. It doesn't follow from the definition of "purple and yellow striped brontosaurus" that a being greater than a purple and yellow striped brontosaurus cannot be conceived. However, it does follow from my argument, where "God" is defined as the being than which none greater can be conceived (obviously you cannot conceive of a being greater than the being than which none greater can be conceived).