If God is an infinite being (per Scotus, say), and if no finite number of steps in an argument is adequate to the scope of the divine majesty, then the strictures of monadic theism aside (God as a unit axiom, if you will), should attempts to prove the existence of God involve infinitary logic? It seems rather contrary to divine simplicity to suppose that some random line of reasoning 5 steps in length, say, would map to as august a power as the divine nature is said to be, and as long as the cardinality of an infinite argument were self-cofinal, our sense of divine unity would be well-preserved.
Is there any known attempt to prove God via infinitary logic? I don't know that even Gödel's is quite like that, though.
Qualifications: pursuant to Conifold's observations regarding absolute infinity in this context, I would have to recommend using a specific infinitary logic, one with arbitrarily infinitely many quantifiers and arbitrarily infinitely long conjunctions/disjunctions, to wit the logic ℒ(∞,∞). The above-linked SEP article says that this symbolism covers a proper class (and the SEP article on infinity in general, uses the lemniscate for a proper class also). For this to be "fitting," though, the consideration raised in Horsten would have to be brought in, and pressed rather hard. I am not too fond of the proper-class approach to absolute infinity; my issue is not so much that the set/class distinction is an ad hoc way to resolve some of the relevant issues (it might be, but the higher and deeper we go in this realm, the more the ad hoc/not ad hoc distinction itself begins to blur and dissolve), but that it is too easy to verify the well-ordering lemma in this connection (by holding fast what would seem obvious, that |ORD| = CARD). So to reformulate the question: would the proper logic, here, be ℒ(God, God), if God alone is held to be the true exemplar of the absolutely infinite?