Abusing infinity and a dis-proof of some people's god's attributes.
First and foremost, infinity is a concept more akin to an operator than anything else. It has more in common with the symbol "+" than, say, "π." In most cases when someone says something is infinite, what they actually mean is transfinite. The difference can be represented in Pascal's classic example of greater and greater amounts of uncountable sets. (An uncountable set is a set for which every member's existence implies another member, such as the number of digits in π or the enumeration of the primes.)
1. Create an ideal circle by integrating progressively more sides into a polygon.
2. Between every side of a polygon is a vertice through which a ray may be extended from the center of the shape.
3. An ideal circle is merely a shape with an uncountably large number of vertices.
4. If we could extend rays from the center of the ideal circle to every vertice joining the sides, the inside of the circle would seem to be solid.
5. Any two rays projected from the center of a shape through its vertices diverge proportionately to the distance at any point from the exterior of the shape.
6. Thus there are not enough rays extending from the interior of the shape to fill the outside area.
7. A larger ideal circle created around the original will require more rays for every vertice than the inner circle.
As a side note, we can once again loosely disprove some attributes of gods by extending this reasoning just a bit.
8. If the overall information representing a given diety is ennumerated by a set of vertices then the vertices may be arranged in a circle.
9. A similar figure may be constructed as a sphere, with corresponding ennumerated values of information.
10. Even though the number of vertices in this circle is uncountable there will never be enough rays to attach the spherical projection to the circular projection.
11. By certain definitions, "god is infinite."
12. By certain definitions, "god is unchanging"
13. The god described by the sphere must then be the true god.
14. There are still more dimensions in which to produce figures with an uncountable number of sides.
15. A sphere produced in the next dimension would be called a hyper-sphere.
16. Again, we must infer that this new figure must be the actual god.
17. The set of propositions which could continue this way is uncountably large.
18. Thus, god cannot be both unchanging and infinite, as infinity itself implies change.
Perhaps the way infinity actually works is a little more clear. As I mentioned, when we say the number of prime numbers is infinite it is taken to mean uncountable. This is because every prime number's existence implies that there is another prime after it. Filip Saidak's proof shows this relatively simply:
1. Let n > 1 be a positive integer.
2. Since n and n+1 are consecutive integers, they must be coprime, and hence the number N2 = n(n + 1) must have at least two different prime factors.
3. Since the integers n(n+1) and n(n+1)+1 are consecutive, and therefore coprime, the number N3 = n(n + 1)[n(n + 1) + 1] must have at least 3 different prime factors.
4. This can be continued indefinitely.
Now we can address the notion that a real number "divided by infinity" equates to zero. What is usually meant by this notation is that the limit of a real number divided by another number approaches infinity as the divisor increases. Zeno's paradox of movement presents a problem with this. If something attempts to move from any point to another it must, in an ideal universe, traverse an infinite amount of distance. That is, the distance moved divided by the distance remaining. Not matter how it is arranged, the result will always require more movement. We now know that there is a basic "pixel" for reality which forms the basis of quantum mechanics. This is known as the Planck unit, defined in terms of the five physical constants. It forms the basic unit of measurement for everything, being composed purely of observation of physical laws. So it is that we could not even move without some simple, self-evident axioms.
To claim an "infinite" god exists is to undermine the very observations we make that allow us to function, as well as the existence of the god itself. Such a thing is contradictory to the existence of our universe. Logic and god cannot co-exist, the universe cannot exist without logic, and God cannot co-exist with the universe. Logic is more than simply a contrived human notion. It is integral to reality, as it is a formalization of the way reality works regardless of how we think and feel about it.