- (Premise) There is no set of all true propositions.
- (Premise) For any kind K, there is a set of all actual entities of kind K.
- (Premise) Divine believing is a kind.
- (Premise) God believes every true proposition and no other propositions.
- (Premise) If there are at least as many Fs as there are Gs, and there is a set of all of Fs, then there is a set of all Gs.
- There is a set of all divine believings. (by 2 and 3)
- If there are at least as many divine believings as there are divinely believed propositions, then there is a set of all divinely believed propositions. (by 5 and 6)
- If there are at least as many divine believings as there are divinely believed propositions, then there is a set of all true propositions. (4 and 7)
- It is not the case that there are at least as many divine believings as there are divinely believed propositions. (1 and 8)
It is natural to think that when x believes p and believes q, then there exist two believings. But by (9), this is false in the case of God. How could that be? Maybe when God believes a proposition p that obviously (to God) entails a proposition q, his believing of q is not a distinct believing from his believing of p: he believes q by believing p. After all, it would have been inaccurate for you to have said five minutes ago: "Alex does not believe that it is true that 82=64." But prior to thinking about this, while I did believe that 82=64, I had no separate belief that this proposition is true. Nonetheless, arguably, it was appropriate to credit me with the second-order belief that it's true that 82=64. Perhaps, then, God has only one act of believing, where he believes a maximally specific true proposition that obviously (to him—all entailments are obvious to him) entails all truths.
And if divine simplicity is true, then that one divine act of believing is identical with God and has its content extrinsically.
I think the most disputable premise in the argument is 2. What if there is a proper class of Ks, say? But the argument can be re-run with proper classes in the place of sets. One just strengthens 1 to say that there is no proper class of all true propositions. And that is correct, since for every proper class C, there is a true proposition that C=C, and there is no proper class of all proper classes. So even if one replaces "sets" with "proper classes" in 2, the argument can be run. And it is plausible that there will be no replacement that will do the job. For no matter what kind of group-type G we put in place of "set" in 2, we will probably be able to run some analogue of Cantor's diagonal argument to show that there is no G of Gs.
What if we apply the argument to the kind true proposition? I think this is perfectly legitimate and shows that propositions aren't actual entities. Facts like <There are infinitely many propositions> aren't grounded in infinitely many propositions, but in something else, like the unified divine nature.