Here's some logic (and lets be clear: I'm saying that the christian god cannot exist as defined by common christian belief within the bounds of logic).

Problem of Omnipotence: If God is omnipotent, he could create something even He couldn't solve or do. But if he couldn't, he was never omnipotent in the first place.

Problem from Free Will: God is a personal being, and personal beings are said to have free will. But if God knows everything, he already knows his decisions, so he doesn't have free will. So, he can't be all-knowing if he is to be a personal being, and therefore cannot exist.

A benevolent all-knowing being only has one option in any situation, and that is the option that causes most good. Therefore a perfectly good all-knowing God has no free will. An all-knowing god instantly knows all of its future actions and therefore has no free will to change them. A god with no free will is not moral. If an all-powerful and all-knowing God exists then this (by a long chain of cause and effect) denies any free will of any living being. If God has free will, but never chooses evil, it is immoral because it could have created life in the same way: With free will, but also never choosing evil. Therefore God must be immoral, not all-powerful or not all-knowing.

Existence is a relatively simple concept -- it is defined as that which consists of either matter or energy. Therefore if a god exists, it must be composed of either matter or energy. If I argue that something exists, but then claim there is no way to detect it, my argument contradicts itself. Let's say I tell a deaf man that I hear a deep, loud sound coming from a speaker. If he lays his hand on it and feels no vibrations, he has every right to be skeptical. If I say that this loud sound does not have vibrations, he may then pull out his trusty microphone or other sound wave detector. If this instrument detects no sound in the vicinity, can I still tell him that this loud sound is occurring? At some point, if my definition of "loud sound" basically boils down to "that which is the opposite of any evidence that a loud sound is occurring," then clearly my approach to truth needs a little work.