First, on acquiring a familiarity with Gödel's modal proof (his ontological argument), the following paper is an absolute must read:

"Benzmüller, Christoph, and Bruno Woltzenlogel Paleo, An Object-Logic Explanation for the Inconsistency in Gödel’s Ontological Theory, 2016."

I reproduce a simple result:, one statement from that paper that may refute it and lend greater credence to Gödel's proof.

"As a model for P(N(⊥)) (which is a consequence of Gödel’s axioms) cannot be built, Gödel’s axioms are inconsistent."

I simply state that the finite field GF(8) (and its child GF(4)) and the metaphysical model made of every such disjunction surely suffices as just such a model also built on the definition of the Godlike essence, and so the modal proof is properly recovered from it. The model includes that of the empty essence claimed to lead to P(N(⊥)) as only placed on one side and never in the empty middle of every disjunction with but one side extant: that empty essence a consequence of the ambiguity of Gödel's single principal element - in truth found to be separate elements on each extant side of the freely decided disjunction(s) found between such disjoint sets.

The statement P(N(⊥)) is equivalent to the statement that no possible world is accessible from that "world" of falsity as it is completely inconsistent and the "accessibility relation" may only be reflexive, if that. (Yet then completely inconsistent).

Then the statement φ ∨ φ is a nonsense, as no world on the left can access the world on the right even if it remains inconsisent; if φ is then the empty self-difference, then the statement is only reconcilable to φ = φ, a contradiction; for the accessiblity relation cannot so apply in the middle across that which are surely in disjunction as possible worlds. For any self-difference then, I must have some statement of Ω ∨ φ instead.

Then such a disjunction of φ ∨ φ leading only to ⊥ ∨ ⊥ brings a contradiction in itself rather than inconsistency: that an empty essence may apply as an essence for a composite individual partitioning Ω into "incompossible" sets within disjunction.

Then it is impossible, for two worlds that are "non-compossible" and held in disjunction to share the same essence, and so the empty essence must never apply to one of the two (in fact both). (Disjunctions must be necessarily decided one way if Ω ∨ φ is ever the case.) The statement is not so much as to whether the empty essence may apply both sides and to lead to complete inconsistency overall, but the statement is actually, "Is there a model with Gödel's axioms that will not lead to such an inconsistency?"

This section of the site states that there is, and it that it never will. To assume the model in GF(8) first "begs the question" I also admit - but this model is not only a good fit, it is pleasing also.

Gödel's proof is found correct if restricted to an ultrafilter in positive predicates: but that there is more than one ultrafilter (the seven of which are also disjoint) has not been adequately accounted for. As there is no middle between these sets and Gödel's axioms state each ultrafilter of positive predicates is formed of compossible predicates, the set of all positive properties is actually able to be partitioned. Middles of freely decided disjunctions formed between these ultrafilters are then "empty" in predicates between these same sets in the partition: That "false" theorem may only find it's place on one side, and never in the empty middle then logically "inverted".

As each instance of these filters are separately in modal collapse (each the result of a free application of divine liberty), disjunctions are, by axiom, just as necessary to that partition as an overarching "God" is to the concept of any positive essence or other positive predicate. That incompatible "falsehood" of the empty essence φ supposedly leading to P(N(⊥)) then remains necessary to every disjunction and partition, and then also necessary to God as one side is freely chosen; the other becomes inaccessible; God is forever consistent.

Toward proof I offer the model of the trinity in my book "Seven Eyes Open" (fourth edition). The statement P(N(⊥)) relies on no possible world being attainable/reachable from the one in which N(⊥), or rather N(φ) is provable. If that "empty essence" φ<=>⊥ is such that every middle is found necessary to the partition, then "every possible world" (where P(x) holds for every positive predicate so possibly composed or conjoined together) consequent from every application of free virtue made in every divine choice (as of every possible partition in Ω) are also accessible worlds and possible ones also, so that the model with GF(8) has indeed been built without antinomy - in reaching that of P(N(⊥)).

Consider then the disjunction r∨s. If the right side is extant, the disjunction is no longer r&er∨p&ep&r-1=>s&es but instead God's essence er must then exit on the right side, so forming: r&er-1∨p&ep&r-1&er=>s&es, and so that essence er is then found exeunt upon the right side also.

Then the left side is discarded as never extant and there is effectively the empty essence that side; only referencing the middle in self-difference with a closed ultrafilter in all s, whilst the principal elements operate to give ep&er = es to close upon that side extant, and to give that ultrafilter in all s it's own essence. Such "empty" essences are necessary to the partition; Gödel's ultrafilter (i.e. any arbitrary one of the seven) appears to remain completely intact whereas the empty essence so found never reaches equivalence to ⊥ in the middle, but ⊥ applies only on the one side never chosen extant. As this is the case in every freely decided disjunction (never failing in privation or otherwise in falsehood as ⊥) with those seven principal elements forming GF(8), for every possible world (or with that empty essence for every similar individual - cf. Act 17:28) that middle is always found present and the empty essence lemma is then not an antinomy. By that application (in all such disjunctions) it never leads to ⊥ extant, for that middle - also that same set as the set extant - is an ultrafilter and also closed with a principal element as es above: an essence.

Of course, there are details: only one set in the model GF(8) may ever be considered inconsistent; as any possible world (one side of the disjunction) is decided against. Yet, the only set that may then be found troublesome is the unity - as that of virtue. As that principal element is regenerated from the remaining six elements in C7, or even the elements of GF(8)+ (the octal), the model is strong enough to hold; for the remaining elements absorb the unity: unity is then a special case, as is the zero also if a similar redemption of the model may hold once unity is recovered.

So, here, as in the book I state that Gödel's definition of NE requires extra care, for there is not in truth one singular predicate of NE, but seven predicates each in disjoint sets, forming ultrafilters. Gödel's definition of NE then needs strengthening and the model of the Trinity applied in the form of GF(8), with the operation of the octal made over the principal elements, zero inclusive.

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