Here is how proof from the Godel Theorem shows God exists through super axiom:

Human beings can conceive of an axiom through which complete mathematical intelligibility can be completely formalized and proven – so that all mathematical questions can be correctly answered from within the mathematical parameters of the “super axiom.”

If human beings can conceive of such a super axiom which computers can now show (through Gödel Theorem), must be the ground of all mathematical intelligibility (from the most complex to the most simple), then such mathematical intelligibility must exist so that it can be conceived of by human intellection.

If such a super axiom does exist, it must exist through a mind capable of conceiving it originally. Such a mind would have to be higher than the mathematical super axiom conceived of. This higher mind is God.
This is an improvement over the Anselmian Ontological Argument (which does not work) going back to the 12th century, because it is grounded in Gödel’s Theorem itself. Recall from previous work that Gödel shows that there exists mathematical intelligibility higher than that which is known by human beings at any given time. Human beings leave behind previous sets of axioms, and make a conceptual leap forward into the domain of higher mathematical intelligibility in order to formalize and prove new developments in mathematics that cannot be reduced to previous sets of rules, axioms, and algorithms.
This shows that mathematics is not built up from below – using arithmetic and then making it more complex, but rather exists on a higher intelligible level. Human beings intuit this higher level, and use it – rather than the previous set of algorithms – to ground higher mathematical developments. The fact that human beings can intuit and conceive of this higher mathematical intelligibility shows that it must somehow exist in order to be “available” to be grasped. What Gödel recognizes is that this available intelligibility seems to have preexistence before the first mathematician grasps it and makes it the ground of new mathematical developments.
Gödel’s ontological proof postulates the highest possible mathematical intelligibility – “the super axiom” that can ground the whole of mathematical intelligibility, showing that mathematics is not grounded from the bottom up, but from the topdown much like Plato and Pythagoras had thought. Gödel completes the argument by saying that if the highest mathematical intelligibility exists, it must exist through a higher or greater reality – namely, through the mind that conceived it – the mind of God.
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