Personally, I am glad I do not believe in a God who is controlled by anyone's human logic. If we see a paradox, then perhaps the limitations are ours. Unless you truly believe YOU are omniscient.
Some of the best known philosophers, such as St. Anslem of Canterbury argued the existence of God (often called the ontological argument) with statements such as:
1. God is the entity than which nothing greater can be thought.
2. It is greater to be necessary than not.
3. God must therefore be necessary.
4. Hence, God exists necessarily.
or,
1. God is something than which nothing greater can be thought.
2. God exists in the understanding.
3. It is greater to exist in reality and in the understanding than just in understanding.
4. Therefore, God exists in reality
You might also look to the works of René Descartes. Here is a wonderful quote by him.
Whatever method of proof I use, I am always brought back to the fact that it is only what I clearly and distinctly perceive that completely convinces me. Some of the things I clearly and distinctly perceive are obvious to everyone, while others are discovered only by those who look more closely and investigate more carefully; but once they have been discovered, the latter are judged to be just as certain as the former. In the case of a right-angled triangle, for example, the fact that the square on the hypotenuse is equal to the square on the other two sides is not so readily apparent as the fact that the hypotenuse subtends the largest angle; but once one has seen it, one believes it just as strongly. But as regards to God, if I were not overwhelmed by philosophical prejudices, and if the images of things perceived by the senses did not besiege my thought on every side, I would certainly acknowledge him sooner and more easily than anything else. For what is more manifest than the fact that the supreme being exists, or that God, to whose essence alone existence belongs, exists?
Of course, neither of these is perfect, and and it is a large subject, the source of centuries of debate and the spilling of much ink.
Personally, I am more convinced by the teleological arguments, often classified as arguments by design. These take the form of something like:
(1) Entity
e within nature (or the cosmos, or nature itself) is like specified human artifact
a (e.g., a machine) in relevant respects
R.
(2)
a has
R precisely because it is a product of deliberate design by intelligent human agency.
(3) Like effects typically have like causes (or like explanations, like existence requirements, etc.)
Therefore
(4) It is (highly) probable that
e has
R precisely because it too is a product of deliberate design by intelligent, relevantly human-like agency.
The discussion gets much more complicated from here. But this will get you started if you really want to. If you are looking for something quick and easy, it is not going to happen.
IMHO, in observing this world with my human mind, I believe it is vastly more probable that God exists than not. That is as close to proving God as logic will ever take us. The rest is faith, but it is not, for me, a faith that is contrary to reason. Rather, reason simply gives us a springboard for talking about the things of faith. This is theology, "faith seeking understanding." But now we are into a different argument for a different discussion.

Kristi