St. Anselm's ontological argument, in its most succinct form, is as follows: "God, by definition, is that for which no greater can be conceived. God exists in the understanding. If God exists in the understanding, we could imagine Him to be greater by existing in reality. Therefore, God must exist." - Wikipedia
This and Godel's argument on existence of God rest crucially on our imagining of God .
The argument that follows rests also on our imagining of God.
It goes as : We imagine God to be qualitatively greater than us so as to be defined as a category different than human; otherwise God would just be a better human in which case we need no new category definitions. Humans can imagine or comprehend infinity but can be safely assumed to be finite*. Thus, God can be qualitatively greater than us if he is infinite. Moreover, if God were finite but still qualitatively greater than us in some other way, then God is not God according to St. Anselm's definition. This contradiction arises precisely because we can conceive infinity (which is greater than finite) if we allow for the fact that conception has both connotations : 1) imagination plus comprehension, 2) creation**. Even better, if God is finite, then even before we conceive infinity to show God down, I believe we can conceive a different finite object that would be greater than God by any definition of "greater" and, more precisely, by the same definition by which God were to be finite. E.g. conceive two such finite Gods or a cake bigger than God. Since only infinity is not greater than two infinities, it seems God needs to be infinite. By the way, this also implies that God needs to be uncountably infinite since if God were countably infinite, we can already conceive objects that are uncountably infinite.
Our present theories and assumptions of physical reality abhor infinity, especially uncountable infinities. Examples : We believe there can not be infinite densities of matter or energy which in my opinion would be an uncountable infinity from a physical point of view. In other words, it is unimaginable how can a possible physical reality correspond to infinite density of matter or energy. The size of the universe can be infinite in principle. More precisely, extensive properties of systems can be infinite given that the universe is infinite. But, we would prefer to have the intensive properties as finite so that we can comprehend physical reality***. We believe physical reality is fundamentally countably discrete, i.e. at the highest energy-shortest length scales, things are discrete precisely to avoid infinite matter-energy densities. This countable discreetness implies that extensive properties can be countably infinite. The degrees of freedom of physical reality at a fundamental level are assumed to be countably infinite similar to the modes on a ideal violin string, but how they exist must be finite in a fundamental way because otherwise we could conceivably arrange them to have infinite densities which should not be possible if our theory can only make sense of finite densities****, unless there are good reasons for the impossibility of such arrangements. Thus, I would conclude that existence of God can not correspond to physical reality unless one can demonstrate a consistent and comprehensible theory of physical reality that allows for intensive variables to assume infinite values. God would be such an intensive property of our physical reality then.
* Remember, we do not consider "imagination or comprehension of infinity" as equal to "infinity". The status of their relation is unclear to me.
** Poetically speaking, imagination too is an act of creation. I am not going to go much down this road since this opens up all kinds of Pandora boxes as to the relation of thoughts and imaginations to physical reality. For our purposes, we assume that when we imagine something, we do not make it thereby to come in to physical existence. As far as physical reality is concerned, when we imagine we - presumably - only make our brain cells change their states. A very sensible assumption in my opinion.
*** This might connect to the logical consistency of physical reality.
**** Can we construct a consistent theory which allows the intensive variable to assume infinite values ?
One example of things going bad is we do not allow dirac-delta functions as legitimate quantum states since they are not squared-normalizable. This is another way of avoiding an infinite density, i.e. we do not want to arrange things so that a system is a dirac-delta function in reality.
