Can Buridan’s formula be rewritten to the following using the standard translation of modal logic?
There exists a world where for all X, X is an element of this world if and only if X is God. Therefore, for all X, there exists a world such that X is an element of this world if and only if X is God.
Buridan's formula can not be rewritten as
There exists a world where for all members x, x is a member of the world iff x is God. Therefore, for all x, there is a world, such that x is a member of the world, iff x is God.
The Buridan formula is a schema that relates modal operators with first-order quantifiers. It is not a simple set-theoretic claim about "being a member of a world". (Possible worlds (in possible world semantics) can not simply be identified with sets of individuals. Doing so, you are basically relying on constant-domain semantics and removing the modal structure.)
The Buridan schema is
◊∀xF(x) → ∀x◊F(x)
If it is possible that all x are F, then for all x, they possibly are F.
It's classical equivalent is
∃x□F(x) → □∃xF(x)
If there is an x that is necessarily F, then it is necessary that some x is F.
If the predicate is read as "is (the) God", this becomes
◊∀x( x = God ) → ∀x◊( x = God )
If it is possible that any x is God, then for any x it is possible that it is God.
You can already see in this natural language "translation", that this is not really a very remarkable claim. (It also doesn't really matter much if you take the predicate as "is the God", meaning, "is the unique x such that x is a god", or as "is a god".)
The Buridan schema quantifies over objects and applies modal operators ("necessarily", "possibly") to a predicate F (or rather: to statements). The proposed rewrite in the post quantifies over worlds and asserts an extensional description of a world's domain (members of the world with some property, in this case the property of "being the God"). You cannot simply "translate" the first into the second. The existence of one special world with some global property does not generally imply that for every object there is some world satisfying a similar biconditional.
If you want to use a predicate that says "object a exists in world w" (E(w, a)) which is to be applied to worlds and objects, then the first part of the proposed rewrite would be
∃w∀x (E(w,x) ↔ x = God)
This is, by itself, a perfectly fine way to express "there is a world in which all members are the God". In other words, it states there is some world with the extension (associated domain of individuals) { x | x = God }. This trivially implies (given normal set theory)
∀x∃w (E(w,x) ↔ x = God)
For all x, there is a world w, such that w contains x iff x is the God
But that proposed rewrite is not an instance of Buridan's schema; it's a much stronger assertion. Whether or not Buridan's schema is valid depends on your semantics for quantifiers, objects, worlds. For instance: Are you assuming that all worlds contain the same domain of individuals (constant-domain semantics) or not? It's not unreasonable to express the statement "a exists" (where 'a' is the name of some individual) as
∃x: x = a
It may not be unreasonable to assume that some individual 'a' did not always exist in (say) our world, or that it only exists in some worlds and not in others. So ∃x: x = a would only be true in those worlds where, in fact, 'a' exists. But the statement
◊¬∃x: x = a
is a contradiction in constant-domain semantics: it's false at every world in every model. In a constant-domain semantics the same domain (set of individuals) is assumed for all worlds. So, you cannot refer by name to any objects that might be missing in some world (nothing is misisng).
It is possible to "translate" the Buridan schema (converse Barcan schema) into a set-theoretic model, but not by simply identifying the modal operator with a statement about objects being a member of the domain of a world. You need a set-theoretic semantics for first-order modal logic, such as Kripke semantics. Part of the semantics will then be an accessibility relation between worlds. It then turns out that the Buridan schema is valid in those Kripke models that have "increasing domains", which essentially means that if you access world w' from world w, all inviduals of word w are also individuals of world w'.
None of this is going to give much or any special insight in "God", however. The main problem with any proofs of God's existence is not that there is something wrong with the logic, or that the argumentations are not sound, or even that the premises are not substantiated by convincing evidence -- in each of those regards there are problems, but the essential problem is that the concept or property of "being a god" (and based on that the concept of "being the unique x such that x is a god") is never clearly defined. Worse than that - even if "being a god" is well defined, and even if it would be provable that a god exists, this does not yet prove that that god is the unique God (or that it is the God that religious people happen to believe in). In religious talk "God" is used (similar) as a proper name. If that makese sense, and you further assume either a constant-domain semantics or that there is really only one possible world (namely, the actual world), then it makes no sense to even ask for a proof of existence: the semantics already presupposes the existenceA.
(A) This can also be defended based on purely theological grounds (in Abrahamic religions). According to the narrative (Exodus 3:14), God when asked what his name is, answered "I am that I am" or "I am who I shall be". This can be interpreted as a refusal to say who he is or as an affirmation of trustworthiness that cannot or should not be second-guessed: a self-asserting, non-contingent existence that cannot be derived from anything else, and thus, also can not be proven.
Long comment
Why you want to remove modalities? without them Buridan's argument loose its interest.
See Plantinga's The Nature of Necessity (1974), page 58.
In a nutshell, Buridan's argument states that God is a necessary being, and thus he lives in every possible world. But is absolutely free: he can refrain himself from creating entities.
Thus, there is a possible world w where God is the only existing entity.
This means that: ◊∀x(x=G).
But this does not imply that ∀x◊(x=G): you and me are not "potential gods".
Your proposal is to "embed" the possible worl" talk into the formula: "There exists a world where for all X, X is an element of this world if and only if X is God", that amounts to: ∃w∀x (x∈w ↔ x=G).
But the inference from ∃w∀x to ∀x∃w is valid, and thus from the formula above we derive: ∀x∃w (x∈w ↔ x=G).
