One argument for God's existence is known as the kalam (Arabic for "eternal") cosmological argument, or KCA:
Explanations and defenses of the kalam cosmological argument (offsite):
Some answered objections to the KCA:
The universe could have always existed
1. Theories that the universe could have always existed are always being proposed, but the majority of the scientific community still holds that the Big Bang theory is the most likely. Since the Big Bang is considered to be a singularity - a point at which the laws of physics can no longer give definitive results - many scientists are agnostic about what could have happened prior to the Big Bang.
While there is a relatively new cyclic universe theory (proposed in April 2002), as Space.com states, "Even the cyclic universe does not address when the cycles began, so 'the problem of explaining the "beginning of time" remains,' the researchers say." (In New Theory of Universe, Time Never Ends by Robert Roy Britt)
For more information:
2. If the universe has always existed, then time has always existed, for matter has always existed and presumably was not entirely inert (i.e. events would occur by which the passing of time could be measured). This implies that an infinite number of events have occurred prior to now, as there was an infinite time prior to now for them to occur. Yet this is the same as claiming that one can count to infinity: it is the claim that infinity has already been counted to, by counting each event prior to now. Since infinity can't be counted to (there is no number at which one reaches "infinity," only an increasingly large finite number), the universe hasn't always existed.
If God made the universe, then who made God?
Atheists object that positing God only raises more questions without answering the original question of the origin of the universe. It only moves the question back a step, so that instead of asking how the universe came into existence, we ask how God came into existence.
However, this objection is true of any proposed explanation for the beginning of the universe, including those proposed by scientists and/or atheists. If the current universe came about through the Big Bang, what caused the Big Bang? If the Big Bang was caused by a Big Crunch, what caused the Big Crunch, and where did all the matter that is contracting and expanding come from?
Actually, positing God does answer some questions which aren't answered by scientific explanations. Positing an immaterial, powerful being as an uncaused cause explains how the physical universe could have come to exist without relying on the existence of matter in some prior universe. And positing God provides a starting point to answer philosophical questions about the purpose of the universe and our lives, which scientific explanations can't address.
See also Who created God?
If there can't be an infinity of time, how can there be an eternal afterlife or an eternally pre-existent God?
There is no problem with there being eternity going forward. Suppose we count each second, starting now and continuing indefinitely. At any point in time a finite number of seconds will have been counted, yet the counting continues forever. There is no contradiction, for we never reach a point where an infinity of time has passed; there is always a finite number of seconds between any future point and now.
One must keep in mind that time as we know it may simply not exist outside the duration of the universe. Perhaps there was a single "timeless moment" before the creation of the physical world, and perhaps there will be another such moment at the end of the world, with no physical events by which we would measure time passing.
See also God, Time and Eternity by William Lane Craig (offsite).
The KCA doesn't prove the existence of God, only an uncaused cause
Atheists object that even if there is an uncaused cause to the universe, it is not necessarily a deity; it could be any imaginable natural or supernatural cause, or even a plurality of deities as opposed to the Christian God.
Proving Christianity is beyond the scope of the KCA; its only intention is to prove (or at least give grounds for believing in) theism. That said, there are reasons to posit an immaterial, powerful being as the uncaused cause.
As far as science has determined, material events have material causes. It seems more likely, then, that an uncaused cause would not be material, but rather immaterial. It's been objected that the first cause of material events would be just as likely to be material itself, since all material events have material causes. However, the fact that we observe material causes of events doesn't exclude the possibility that there are immaterial causes for those events as well. For instance, an immaterial deity could have set in place the laws that govern various cause-and-effect relationships.
The universe is larger than humans can readily comprehend and contains many objects too small for humans to see. Whatever caused the universe had to cause a gigantic mass of matter and/or energy, composed of mere particles, to come into existence. Therefore, the cause must be very powerful.
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