If you want to talk about the "illogical infinite God" you have to explain why an infinite god is illogical.
First, I never said there is anything illogical about infinity, as a mathematical construct. It and infinitesimals are the basis of the calculus. In these cases we are talking about abstractions, not real things, and I think it is an open question whether physical things can be infinite, but that is not the issue here.
What makes an infinite God impossible? This is an old, old theological argument. I tend to express it in modern terms, but maybe you have heard of it as the question of whether God can build a rock so big He cannot move it. If he cannot, then he is not omnipotent; if He can, then He can't move the rock, and, again, He is not omnipotent.
The Roman Catholic answer, usually adopted by others as well, is that God is omnipotent except that He cannot do something illogical. Well, then He is not omnipotent. Of course God cannot do something illogical -- I would say that nothing can do so (illogical in a mathematical sense), but that is not the issue. If God is so limited, then he is only Superman.
Take another issue -- God's reported omniscience. Such a Being, of course, knows the entire future, including everything He will do. I suggest that such a Being is therefore but a machine: He has to carry out His program -- He has no possibility of any freedom.
Another problem with omniscience is how does He know? How can He be sure He knows everything there is to know? He may know an infinite number of things, which might be compared to knowing everything that is in an infinitely tall stack of books, but there might be another stack somewhere that is completely out of His ken. There is no way for an omniscient Being to know it is omniscient, and hence the contradiction.