Proof Of Gödel's Argument
So, Gödel's axioms permit the set of all positive predicates to be partitioned. Then I proceed as follows.
Of the seven sets in the octal, the empty essence lemma may only apply to one set of seven: as the remainder together are incompossible unless that element chosen "common" to all seven ultrafilters is strictly without any possible aspiration to any greater (and common) closure and it must also then only equal itself! This is true in pairs of Godlike sets taken from the octal. Such a closed identity element compossible with all may only be so "closed" and also self-identical.
Now, if a Godlike set (and ultrafilter) is truly incompossible it leads to an impossible and inconsistent world. Within disjunctions already freely decided (in light of whatsoever is positive and extant); the world not chosen extant attains an empty essence as the Godlike essence of that world never chosen is instead adjoined to that on the other side by the operation of the elements in GF(8) under the axiom of virtue.
At most one set attains an empty essence as the remainder are always incompossible together. The other six must remain necessarily existent, and by symmetry of choice, the empty essence is recovered by the axiom of virtue and the operation of GF(8) embedded in the set of all positive predicates.
The remaining elements are necessary and the operation with them. This holds to the unity in C7 as well as in the octal; for C7 independently retains the virtue in unity without any concern that both unity and zero may be missing from the octal; and both are easily recovered; for they exist together in GF(8) simply because the remainder of the six are always, forever, N.E.
So, on to proof.
In any freely decided disjunction, the predicates of N.E. - the principal elements - exit one side by axiom of virtue. If but only one of the seven Godlike sets' essences may ever be replaced with an empty essence, every disjunction deciding upon that set is "forcibly" decided. This occurs in those several disjunctions but for the sets which act as virtue by axiom of virtue, which must certainly hold in the model with GF(8). For if that axiom holds there is no applying an empty essence to the set overall; the axiom of virtue must also apply in the minimal case where, say, the disjunction is as e_{r}∨e_{s} instead; generating the principal element of e_{p}. Then every set is recovered but for that one set in the disjunctions so necessarily decided, one of r, s, u^{-1} and v^{-1}. The only set(s) not acting as virtue by axiom of virtue (and as uniquely so) are those four in the disjunction itself, which does not include those in the K4 form of the model, i.e. (r&s)^{-1} and u&v (which uniquely act as virtue also, and therefore the K4 form is always present by axiom of virtue in its own Manifold of Virtue).
As the principal elements are always regenerated by products in GF(8), for any individual in the Godhead, there is no empty essence, which was only free to apply but to just one of the seven sets in the octal.
Then if I assume the axiom of virtue; the empty essence is not compossible with all (in fact any partition within the model then freely chosen by liberty - an argument against modal collapse, (see the book "Seven Eyes Open" (Fourth Edition))) such Godlike sets of positive predicates; therefore I have not thrown out Godel's definition of an essence at all; rather, I have proved something; that there is no empty essence on a particular Godlike being, as with the model of GF(8) taken from the "Revelation of Jesus Christ" from the KJV.
Yet if I assume the empty difference to be compossible with every essence of a Godlike being; I reach an inconsistency; a contradiction that God could ever possibly have such an empty essence allowed by Gödel's own definition. Then without the axiom of virtue, as Gödel's axioms allow this; his axioms are not actually inconsistent; merely there is no valid model of any such "God" without adding an axiom such as the axiom of virtue.
The difference is simply the assumption that all positive properties are compossible when one does take into account the "accidental structure of the world." (If indeed God ever has accidents or makes them! - I.e. He is completely consistent; the result may simply be chaotic.)
It is the case that no essences are ever "missing"; the decision(s) between worlds must be found somewhat continuous. (There may well be a calculus over them.)
Then Godel's axioms are completely consistent; his assumptions were simply incorrect. //
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