Godel was a Christian, so he would not be able to see how his theorems disprove God's Existence. Cantor was told by the church that his ever rising infinities usurp God's Almighty power, so Cantor developed the Absolute Infinity--an infinity that surpasses all other infinities. The Absolute Infinity is the one that God has. However, this Absolute Infinity has been shown to be just another set and can be surpassed just like any other infinity can.
So, the God System that He uses is some kind of truth ascertaining system. As such, it is subject to Godel's Incompleteness Theorems. God doesn't have a Meta God System to check the validity of the God System because that Meta God System is also subject to the Godel limitations within itself. There's not a Meta Meta God System to check the Meta God System etc. The God System itself we can suppose is all there is, and it is limited by Godel's proofs.
I do not see error in my thought.
You can prove something not there exists.
Assumption: The empty set exists.
Claim: The empty set exists.
Proof: By assumption.

This trick doesn't doesn't work for God.
Assumption: God exists.
Claim: God exists.
Proof: By assumption.
It doesn't work because God isn't nothing. God was imbued with properties (omnipotence and omniscience to name two). These properties can be shown to be impossible as I do in my writings.