If I were to prove that someone exists, I would have to prove his/her existence along with every one the characteristics that make up who he/she is. To fail at doing so would not provide complete proof. Lets say a group of children are born and spend their growing up years, even have children of their own and grow old, never seeing the world outside the cave. To explain to such a person the concept of the sky would require teaching him/her about the simplest fundamental elements of the sky. It surrounds the earth, holds air in place, is too far away to touch, looks blue, etc You cannot accurately describe a tree without wood, or snow without mentioning that its white. Nor can you prove God without including the mystery. The existence of God should not be proven because to prove that God exists, one would have to take away the mystery. Mystery being an essential characteristic of God, to prove that He exists would simultaneously prove that He doesnt, and since he does, it is impossible. In 1 Timothy 1:17, Paul provides three characteristics, imperative to the nature of God. Without any of these three, God would not be the God I believe Him to be. These three attributes are eternal, immortal, and invisible.

God must be eternal. This is a mystery to the human mind because we are held within the bonds of time. We cannot live, nor even think, outside its realm. This timeless God is beyond us, but also somehow involved in our time-bound lives.

God must be immortal. He possesses both supreme power and permanent authority. This, like Gods eternal nature, this is a mystery to mankind because it is something we cannot possibly attain. The source of life itself, He is constantly creating and renewing the created. This can't make sense. The sun is our source of light and heat, but scientists agree that it will one day cease to function. God's sustaining life, on the other hand, will never run out. This also adds to the mystery.

God must be invisible. He has made Himself available to us in the form of the son, Jesus. The Apostle Paul called Christ the image of the invisible God. So, through His awesome power, the invisible God made Himself available for a time, but no one has ever seen Him. Jesus Himself said, "No one has ever seen God, but [I have] made him known." This is a profound mystery to a people that place so much value on what can be touched and seen.

There is an underlying theme of mystery behind these three fundamental characteristics. Mystery is such a part of Gods character, it must be present. Without wood, you have no tree. Without mystery, there would be no God. Therefore, His existence cannot be proven. Granted, there is incredible evidence, but no undeniable proof.