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God knows everything:
For God is greater than our hearts, and he knows everything.
But He chooses to forget one thing:
I, even I, am he who blots out your transgressions, for my own sake, and remembers your sins no more.
Therefore, God knows everything except that which He chooses to forget.
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The Gospels make it pretty clear that omniscience is a trait that God alone enjoys. Consider the science behind omniscience. He literally knows when the neurons fire in the brains of people all over the world.
Jesus would not trust himself to them, for he knew all men
John 2:24
Come and see a man who told me everything I ever did! Could he possibly be the Messiah?
John 4:29
Knowing their thoughts, Jesus said, "Why do you entertain evil thoughts in your hearts?
Matthew 9:4
If you're asking whether God knows if a tree falling in the woods makes any sound:
Who makes the rain fall on barren land, in a desert where no one lives?
Job 38:26
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The Bible says God is perfect in knowledge, knows all thoughts, all secrets, sees all and no-one can hide from God.
Talk no more so very proudly; Let no arrogance come from your mouth, For the LORD is the God of knowledge; And by Him actions are weighed. 1 Samuel 2:3
for he views the ends of the earth and sees everything under the heavens. Job 28:24
Do you know how the clouds are balanced, Those wondrous works of Him who is perfect in knowledge? Job 37:16
I know that you can do all things; no plan of yours can be thwarted. Job 42:2
would not God have discovered it, since he knows the secrets of the heart? Psalm 44:2
For there is not a word on my tongue, But behold, O LORD, You know it altogether Psalm 139:4,7-8
Where can I go from Your Spirit? Or where can I flee from Your presence? 8 If I ascend into heaven, You are there; If I make my bed in hell, behold, You are there.Psalm 139:7-8
Great is our Lord, and mighty in power; His understanding is infinite.Psalm 147:5
The eyes of the LORD are in every place, Keeping watch on the evil and the good. Proverbs 15:3
For My eyes are on all their ways; they are not hidden from My face, nor is their iniquity hidden from My eyes. Jeremiah 16:17
Can anyone hide himself in secret places, So I shall not see him?” says the LORD; “ Do I not fill heaven and earth?” says the LORD. Jeremiah 23:24
And they prayed and said, “You, O Lord, who know the hearts of all, show which of these two You have chosen Acts 1:24
And there is no creature hidden from His sight, but all things are naked and open to the eyes of Him to whom we must give account. Hebrews 4:13
And by this we know[a] that we are of the truth, and shall assure our hearts before Him. 20 For if our heart condemns us, God is greater than our heart, and knows all things 1 John 3:19-20
There are some epistemological problems with an omniscient god, but I don't know if it's in the scope of this site and it certainly is not written in the bible.