Objections Enumerated

I should enumerate my objection(s), so I list them as follows:

Primary Objections:

  1. Positive properties are compossible, but not across disjoint ultrafilters.
  2. Across a disjunction, the empty essence does indeed "bind" a variable in its scope, that found of the opposing ultrafilter, with its own separate essence of N.E.. The result is incompossible. Under Scott's axioms, the empty essence certainly holds predicated on the one side not extant (on the left side as above in all "r"), as its principal element (its essence) is instead (in the operation of GF(8)+) enjoined to the one side extant; leaving the essence of that other side in φ empty. Under Gödel's original axioms, the empty essence is free to apply either side (in fact both) without predication (as by Scott's conjunct) and is also free to hold open, maximised, to the side which is extant - yet must be predicated to the remaining side empty (due to antinomy); the essence once again leaving that set's essence free to enjoin to the other side - as it is so found necessary in the operation of GF(8). It is also free to appear, then, to apply to the middle as objected to before. Then the middle is the statement that x = y, concerning the being which is necessarily existent - an incompossible set in positive properties, only met in the Godhead's octal group formed in principal elements closed - and without further aspiration (see later) but for modal collapse.
  3. N.E. (as by Gödel’s definition) states all essences must entail N.E., but it is not clear concerning separate or composite individuals forcing a partition of Ω into disjoint ultrafilters. Two such individuals opposed in disjunction cannot be both found of empty essences, else the world (disjunction) is impossibly empty and inaccessible or the disjunction is equivalently decided both sides. Any non-empty middles are inverted negative instead. The ultrafilters either side are found disjoint and closed in their own "compossible" positive properties so legitimately "composed".
  4. If a disjunction has empty essences both sides the world is certainly empty; no such world has any model, unless it be purely reflexive, and also connected to a possible world as above by the empty essence and axiom of virtue, for a non-empty world on the other side of the disjunction and one which is (of necessity) decided upon, leaving that with the empty essence inaccessible.
  5. The Godhead is completely consistent in all positive properties, never failing in privation. There is no reversing any divine act and so worlds decided against become completely inaccessible. The consistency in the octal is forever upheld. God, cannot be tempted. The result is one partition of Ω into disjunctions that never fail in privation and there is always the axiom of virtue.
  6. The definition of N.E. itself does not identify x holding φ ess x with y having N.E., merely that there is such an individual y, and as it turns out here, one that cannot bind the one found on the other side of the disjunction that then necessarily exists in a separate ultrafilter incompossible with that which holds the empty essence. Thus, the composite individual in GF(8) regenerating any essence in symmetry thereby, also fulfills the same definition of N.E. whilst opposing the empty essence, by exiting one side only in a minimal disjunction formed by axiom of virtue. (A binding of virtue present in every disjunction upon the principal elements.)
  7. So, whilst Gödel’s axioms are indeed unsound for “monotheistic ultrafilters” with a singular instance of N.E., they need not lead to any inconsistency in GF(8). In fact, they cannot do so in GF(8) by inspection of the operation and octal’s schema, and Gödel’s proof is recovered.
  8. So, a model exists in GF(8) because Gödel’s axioms are found consistent, but they merely lack such an axiom of virtue.
  9. Dana Scott’s axioms lead to modal collapse but no inconsistency by application of the axiom of virtue.
  10. Gödel’s manuscript states any two essences are necessarily equivalent, but this does not follow for disjoint and incompossible ultrafilters not able to be so composed; for it is a fault that Gödel’s argument did not hold to any sense of the axiom of virtue.
  11. Gödel’s manuscript states that his definition held positive properties to be "independent of the accidental structure of the world"; he had not considered that "solving" for virtue within disjunctions could also hold firm the very same definition of a positive property by simple "binding" (axiom of virtue). Ultrafilters that are morally or otherwise purely aesthetic have a limit of maximality (by positive aspiration, as by Zorn's lemma) and so no negative (incompossible and also "negative" (as necessarily disjoint sets found conjoined or "composed") within the partition) proposition is ever found by aspiration; but only the result of virtues bound to negative statements entailing only the positive. I.e., a negative predicate may (as bound with a virtue) entail a product of positivity also closed as an ultrafilter, a disjoint set closed in its own completeness. As a positive property only entails positivity this does not stop a negative property entailing positive properties by axiom of virtue; such disjoint ultrafilters so entailed remain closed.
  12. By aspiration only in the possible and non-negative, Gödel's definition of "positive property" and also "Godlike" therefore permits disjoint and incompossible ultrafilters with the axiom of virtue.
  13. So, I will find a contradiction as to whether all positive predicates are compossible, for the empty essence φ therefore, surely, is not! This leads to the axiom of virtue freely deciding all disjunctions, for if the empty essence is applied to one side the disjunction is always (or has been) freely decided to the other. (Virtue is free to rest on the middle instead.)
  14. Ω (the set of all positive predicates) partitions due to the accidental structure of the world; essences are no longer necessarily equivalent but they do remain necessary. Gödel’s assumptions are wrong but his axioms, on including that of virtue, are found to be completely valid as an independent axiom only requires adding. I otherwise reach a simple contradiction that proves the empty essence φ is not a compossible property.
  15. I must also state the absence of any "divine dipole" shared in the symmetries of the octal with positive predicates: God has simply chosen always to operate in positive properties without privation, continuing in virtue, as in the didactic. His symmetries allow Him the choice of operation in singletons, whereas the octal schema in which they move is unaltered. Once a free decision is made, it is never reversed, the world is not rewritten as to ever reverse to undo an act - as that world is forever inaccessible with an empty essence, if any at all - if it may be said to exist at all - it is as inaccessible as is ⊥.
  16. So, I conclude that Gödel’s axioms are found completely valid, requiring but one further addition to his original axioms! (His assumptions are invalid, the axioms are valid.)
  17. Gödel’s first definition as of the "Godlike" predicate appears the only troublesome axiom (regarding (in)compossible ultrafilters) but it is not so. It is reliant on the closure of each ultrafilter, for to ascend any further than closure on one side of a disjunction in virtue (as by that axiom of virtue) entails a privation on the positivity of that predicate (and ultrafilter) exeunt on the other side, and therefore to ascend further attains or aspires to a predicate which is not a positive property as by Gödel's first axiom. Such negative properties do not follow from the Godlike and Ω must remain partitioned! Each essence of the seven fully composed in the possible positive predicates is then found Godlike, that troublesome axiom and definition of "God-like" remains valid, as no "positive property" can break the closure of the ultrafilter by any means of aspiration. (It may not do so, it is impossible to break the binding of virtue without also entailing a negative result, and then that is not a positive property. Ultrafilters are so closed.) The individual x in the essence G(x) is a-priori composite to begin: Gödel's definitions and axioms remain valid.
  18. So, what is a positive property? Simply that equivalent or set completed which will entail its very own N.E. in the Godhead. Positive predicates will not conjoin positive if that requires unbinding a virtue or equivalently breaking a sets closure. The middle will always remain "empty", non-exemplified.
  19. The empty essence is then applicable to every individual as every freely decided disjunction may surely exit one side, but it remains incompossible with any genuine (non-empty) Godlike essence. No ultrafilter ever contains the empty set: (There are also no possible empty worlds, there must be an essence in every such ultrafilter.
  20. There is then permitted equality between the set of virtue(s) acting on a privation and the resultant positive set / product with N.E. It remains that this set is closed positive with no negative statement reached by axiom of virtue, as by aspiration or Zorn's lemma within the product of virtue(s) bound. Virtue remains a closed set, and the result is then, in kind, one also closed yet without that limit discoverable. Every positive property in the result is a result of binding, yet it, as a set, is found disjoint to all others in the partition of Ω.

Then all of Gödel's axioms are upheld valid, but there is one further axiom (that of virtue) along with its equivalents that state there are "products" of binding that lead to a partition of Ω into disjoint sets. Modal collapse, then, is found no problem also. (See the book "Seven Eyes Open" (Fourth Edition).)

So, by adding an axiom (the axiom of virtue) Gödel's proof follows complete: his original (and also Scott's) axioms needing no modification at all. I, by virtue, simply reach a result that the empty essence is not a compossible filter; yet there is found a model for it; a disjunction in virtue having exited one side only. Then that empty world is never extant, and as an empty world it also has its own place: as excluded from every ultrafilter. It then belongs only in an empty world (but for any non-empty essence that must have departed from it in free choice leading to an equally possible world so found extant).

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