# Should proofs of God involve the infinitary language ℒ(∞,∞)?

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[16] 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?

• Since the length of a proof depends entirely on the axioms, I can imagine no justification for the suggestion that there is any connection between length of proof and "divine majesty". Oct 25, 2023 at 19:05
• One problem about supposed intellectual or evidentiary proof of 'God' is that the believer will be even more inclined to make value judgements about non-believers. If you believe in 'God' you should have no need to prove it. How defensive is that? Oct 25, 2023 at 20:23
• As we know from Cantor, the divine nature of the august power inheres in the inexplicable Absolute Infinite that all of the transfinite does not even touch. Before the Absolute everything is zero, as they say, so infinitary logic is no better than ordinary one in this regard. Remembering the fate of the tower of Babel, we better stick to what fits our nature in its finitude, as intended by the Divine, so as not to offend the august power with our infinite hubris. Oct 25, 2023 at 23:13
• I don't see how an infinitary proof would really make any difference. Many traditional arguments such as the uncaused cause, or the unmoved mover, or the absolute noncontingent are in effect arguments by reductio. They are disputed because their premises are difficult to verify, not because there is a problem with the logic as such. Oct 26, 2023 at 3:58
• Even for well behaved finite quantifier infinitary logics you need weakly κ-compactness to be necessary for κ-completeness to be able to prove any truth entailed from any weakly κ set of evidences where it's assumed that κ≥ω1, which inevitably leads to the axiom of weakly compact inaccessible cardinals that cannot be proved from the usual axioms of any set theory. But such weakly inaccessible cardinals intuitively and provably contradict Gödel's famous axiom of constructibility thus even if you can prove said entailed truths as facets of the oneness, V cannot be constructed from the bottom... Oct 26, 2023 at 7:35