The Godlike And The Necessary
Godel's definition of an "essence" was of a positive predicate that entails every other possibly positive predicate, and that includes necessary existence. Across all possible worlds, it becomes possible that every predicate may be positively exemplified or not so, entailing a modal collapse, where every predicate or its apparent negation is necessary: Of course, one is positive and its negation not so, but there is no sense of liberty for the statement(s):
Pos(a&b)=>Pos(a)&Pos(b) which follows from "a&b=>a" and "a&b=>b"
Is met also with the statement that Pos(a)&Pos(b)=>Pos(a&b) from the properties of an ultrafilter.
However the modus tollens, ¬b=>¬(a&b), in some world entails Pos(¬b)=>Pos(¬(a&b)) etc.
Yet if Pos(a) is true, the last above can not be well formed unless every statement is necessary (and from some principal element (necessary existence), the same essence follows as the conjunction of every positive property). Then, essence is itself also necessary existence and as every predicate is necessary or not, I find ¬Pos(a&b) => ¬Pos(a)&¬Pos(b).
I.e. if ¬Pos(a)=>¬Pos(a&b) I can entail ¬a=>¬(a&b)=>¬a if the distributive law holds ¬(a&b)=>¬a&¬b. This I would assume from the perfect: only positive predicates are exemplified from an essence, and each predicate is positive or not.
So if I further assume from a&b=>a that Pos(¬a)=>Pos(¬(a&b))=>Pos(¬a) from an ultrafilter, then following from Pos(¬a) I may infer Pos(a)<=>Pos(a&b) If Pos(x)<=>¬Pos(¬x).
And then every statement following from that essence (conjunction in every positive property) is a logical equivalent, and every statement positive or not is necessary.
However, in the octal things are quite different. Godel's argument takes pride of place but there is no modal collapse unless there is no liberty!
I then modify my definition of "God-like" to the following:
"Every set closed as an ultrafilter and which contains necessary existence is completed as "God-like".
Then, there is the possibility of the conjunction Ω\e v e, where "e" is rest as before. (See chapter two in the book.) Then Godel's set from the essence would be as Ω v φ and then there is no liberty. However, the right hand side of Ω\e v e also is "God-like", and closed as an ultrafilter, yet whilst entailing no other positive properties.
Then "e" is also an essence to itself! (It is already necessary.)
The omnipotent however, requires all liberty in L(G) within each disjunction, to entail every positive property. That liberty is also an essence, even to itself. That liberty must be omnipotent in all virtues.
In creation, however, (see chapter sixteen, "Time No Longer II"), if (l v a&l^{-1}=>l) say, then (l^{-1} v a&l=>G) in the octal, and if a<=>l (l is principal in creation) then φ v l=>G, and there is an equivalent modal collapse. (Then, "l" is rewritten as fully "God-like" when free in all positive properties and is still also an essence for itself, though not for God, whose new name in g is instead justified as g<=>e&l.)
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