Suppose that the greatest conceivable being (GCB) exists in the mind alone (and not in reality). This is simply to assume that if an Omnipotent, Omniscient, Omnibenevolent God exists He would exist in the mind only. Existence in reality is greater than existence in the mind alone. Just like one hundred dollars in reality is greater than one hundred dollars in one's mind; because one hundred dollars in one's mind can't buy any Big Mac's. In short, reality is greater than understanding. We can conceive of a GCB that exists in reality as well as in the mind. Every person should be able to get a mental image in their mind of a greatest conceivable being existing in both reality and in their heads. Therefore, there is a being that is greater than the greatest conceivable being. But this is impossible, for it is a contradiction.
It is a contradiction because originally, it was said that the greatest conceivable being exists in the mind alone. If there is a being that is greater than the greatest conceivable being, than it wouldn't be the greatest anymore. Since this is a contradiction, it is false that a GCB exists in the mind alone and not in reality.
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There are many problems with this argument. The first problem, and the most obvious is the language. This argument is poorly worded and confusing. Another problem is that using the same structure provided, one may prove the existence of the perfect island, when we know that it does not exist.
The word "greater" is used throughout this argument. Pay close attention to the tense of the word because it is not greatest, it is greater that is being misused. Particularly in these sentences: "Existence in reality is greater than existence in the mind alone." And "Therefore, there is a being that is greater than the greatest conceivable being." In this context, "greater" is being used like I've never seen it being used before. While it is still comparative, like in other forms of the word I've seen, it seems as if a completely new definition of the word would have to be written for it to be used here.