"proof" is a tricky word. What constitutes ''proof" to one person, may not be convincing to another.
SOME people only consider what they can touch, see, and hear.
Others will accept logic and deductive arguments.
Secondly, somethings can be proven, and others can not. So there are simply 2 classes of things. Those which are known with relative certainty, and those which are not. Both types of things are still known, just 1 is less certain.
So what does this all mean when it comes to proving God? -- it means that God, if he exists, may easily be in the category of things which can not be known with certainty and that there is evidence and arguments FOR his existence, but whether or not they are 'proof' depends on the individual.
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Thirdly, since IF God exists, he is logically metaphysical (beyond the mere Physical)
Science deals only with the Physical, therefore we mostly likely need an additional tool to find evidence for God.
Luckily we have Philosophy and Theology - both which deal in the metaphysical.
So likely, our most convincing evidence for God's existence will be in the form of a Philosophical argument and not empirical scientific theories.
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Fourthly, Too often people combine what I call steps 1 and 2 together. They try to go from God's non-existence to the Christian God, all in 1 step. They are in fact 2 distinct steps. Because you FIRST need to conclude that God DOES exist, before you can argue for or against Christianity. Because logically, if you conclude that God does not exist, then any particular religious philosophy is mute.
Too often Atheists claim they are arguing against the existence of God, but their arguments show that they are arguing against a specific instance of him, and NOT the concept itself.
You ask if God exists -- so you should start there and not try to decide WHICH God - for now.
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Sadly, not many people, now-a-days, are very well versed in philosophy. But none the lesss, here is a link to 20 arguments for the existence of God:
http://www.strangenotions.com/god-exists/
The argument getting the most recent attention is the Kalam variant of the Cosmological Argument.
The Kalam is quite simple, yet powerful:
Premise #1: Things which have a beginning, have a cause
Premise #2: The Universe had a beginning
Premise #3: Therefore the Universe had a cause
The Kalam can then be expanded to show that this cause must be a metaphysical, transcendent eternal, uncaused, mind, which we call God.
For instance, if we expand upon the Kalam, then we can logically deduce that this cause must be, itself, uncaused, or else we need to search for another cause. We know that only transcendent causes could be the cause of the Universe (or multiverse) because if it is not transcendent, then it can not interact with the physical universe. OH, and it goes without saying that this cause must logically be metaphysical, because it can't be part of the Universe which it hasn't created or caused, yet.
RIGHT?
Also.. logically, we know if it weren't eternal, then it, itself, must have had a cause, and therefore can't be the actual cause of the Universe - something else must be. So it has to be eternal if we are to even consider it.
But why a 'mind' ??
The only 2 categories of things which Philosophers know of which are eternal, transcendent, non-physical, and non-contingent are either A) abstract objects like numbers or B) a disembodied mind/conscientiousness
However, we know that abstract objects like the #2 can't 'cause' anything to happen, that's what it means to be 'abstract' -- so that leaves only a transcendent mind. Since minds don't necessarily require any physical body parts to exist (conceptually)
So the Kalam provides us with a rational basis for the belief in God. More study is needed to flesh out other properties of God.