For context, I think this is a powerful, often-overlooked argument for the existence of God, which I find compelling, and potentially completely convincing that God exists. I do also have other arguments that convince me that God exists, so hard to say whether this alone would be sufficient, but it might.
However it is worth discussing some objections, some of which are worth taking seriously and some of which are not.
First, I think we should dismiss any objections which turn on the colloquial meaning of "perfect". I think this is a mistake that philosophical readers might make, but mathematical readers might not. One way to read the proof is that it is saying "There are such things as perfect properties, here are some claims about them". Another reading, the correct reading, is "Here is a category of properties, defined by these axioms, which I am going to label "perfect"". Mathematicians are more likely to read the axioms as a definition, rather than as premises.
I occasionally see objections like "The property of being omniscient or a murderer is not perfect, but is entailed by omniscience, which is perfect, so the entailment axiom fails". That is just a misunderstanding, in this proof, we'd be happy to concede that "being omniscient or being a murderer" is a perfect property.
Another commonly cited misguided objection is one Graham Oppy has made, where the proof can be used to show the existence of multiple Gods. Oppy's claim is that if we take our set of perfect properties, there's another set with all those properties but one, and another almost-God who has all of those properties.
However if we have "constructed" our set of perfect properties by taking all the properties entailed by necessary existence and God-like-ness, then we can't take any perfect properties out of our set without violating the entailment rule. And for people who want to draw on more literature, the classical theist tradition is full of arguments that the properties of God all entail each other, and are in reality only a single property.
I think the questions about S5 are more serious ones, I have a few thoughts on these.
The argument about S5 can be thought of as an accessibility relation question - are all possible worlds which "exist" (in whatever way we think they might) S5 accessible? For S5, the accessibility relation needs to be reflexive, symmetric, and transitive.
We could do a bit of a Proofs and Refutations move here - rather than arguing about whether all worlds are S5-accessible, let's just talk about the subset of worlds which are S5 accessible. We can just restrict ourselves to the S5-accessible worlds! There's at least one of those - this one.
So we could instead just restrict our thinking to all the S5-accessible worlds from this world. Maybe there are some possible worlds which are not S5-accessible, but that doesn't need to bother us. Just specify that when we use [] and <>, we are talking about only S5 accessible worlds.
What tangibly does this mean? Well, it makes the argument a bit weaker - it opens up the possibility that God does exist in *some* possible world, but that world is not S5-acessible to us, so then God doesn't necessarily exist here. Or said differently, "God S5-possibly exists" is a harder claim to establish than "God possibly exists".
But we aren't coming at it directly like that, as you would with Plantinga's argument. Instead we are trying to show a kind of compatibility between necessary existence and "perfect" properties. But if S5-necessary-existence is compatible with the "perfect" properties, then the argument still succeeds, and there's no reason to think that S5-necessary-existence is any more incompatible than just bare necessary existence, whether you take that to be S4 or something else.
I have some other intuition pumps for why S5 is appropriate here, but these are intuition pumps rather than rigorous arguments.
The objects we are dealing with are properties, and God who might or might not exist with some of those properties.
It seems obvious that God is the type of thing which, if possible, is possible in every possible world. If God exists, then God is the ultimate source of all worlds, and so God must at least be possible (and indeed necessary) in all worlds. S5 is the right system to use for discussing God.
What about properties of God? Again, if God possibly has some property, then could there be a world in which God does not possibly have that property? It seems not, as there's nothing in any world which can constrain God from having a particular property, other than Himself. So S5 seems reasonable to use when discussing potential properties of God.
I think this exhausts my thoughts on potential objections to the argument as given, looking forward to hearing what you have to say!
Similar advice is given in Taleb's Antifragile.
A fragile system, with a naughty or mischievous actor inside it, will fall apart. Maybe fragile systems should be encouraged to fall apart.
An antifragile system is one which benefits from a bit of chaos and randomness. Naughty and mischievous actors in these systems are beneficial, injecting the randomness needed for growth and for avoiding decay.
Thank you for your kind words.
I think when we get to this level of reasoning it's less about actual arguments and more about trying to tease out intuitions and develop intuition pumps. Actual arguments are usually the easy part, the hard part is finding an intuition on which to hang an argument!
While I might agree that the metaphysics of S5 is strange, I don't think that's really a function of S5 as much as it is a function of any coherent metaphysics.
For example consider again the accessibility relation over possible worlds. I suspect we all agree it is reflexive, but to deny S5 is to deny that it is transitive or reflexive. I think a possibility relation which is not transitive or reflexive also very weird!
Surely, my intuition says, if world A is possible from B, and B is possible from C, then A is possible from C. Surely, my intuition says, if A is possible from B, then B is possible to A.
I'd have some sympathy for the denial of symmetry if we were talking about a possible future, so maybe you can get to future A from here and future B from here, but you can't get to here from B, or from A, and can't get from A to B or vice versa. Ok, but we are talking about whole worlds rather than future, so I don't think that's the right logic to use here.
Which I think is really my point in the intuitions in my first comment - this is less about whether S5 is true or not, and more about whether it's the right system for the type of objects we are dealing with. Since the type of objects we are dealing with a kind of "universal", my intuition is we should use the most "universal" logic, which is S5.
I read that other thread you referenced and didn't find the arguments particularly compelling, perhaps because I am coming from more of a platonist perspective where I think formal systems exist regardless of what concrete objects exist which might instantiate them. If I didn't think that, I'd likely deny that necessary existence was a coherent concept at all, and so the argument falls apart much earlier!