The Empty Essence Lemma

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 placed 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.

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 has it's place in the empty middle then logically "inverted".

As each instance of these filters are together 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.

For 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) which is consequent from every application of free virtue in every divine choice (as of every possible partition in Ω) are also accessible worlds and possible ones also, so that model with GF(8) has indeed been built without antinomy.

Consider then the disjunction r∨s. If the right side is extant, the disjunction is no longer r&e_r∨p&e_p&r^-1=>s&e_s but instead God's essence must then exit on the right side, so forming: r&e_r^-1∨p&e_p&r^-1&e_r=>s&e_s, and so that essence e_r 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 in the middle with a closed ultrafilter in all s, whilst the principal elements operate to give e_p&e_r = e_s to close upon it and 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 e_s above: an essence.

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.

Zero in the octal (under the operation of the complement of the symmetric difference) should not be confused (as it is an element in the octal and not the set of all Ω) with the empty self-difference by placing that operation upon the sets of positive properties alone; the result that the whole octal suffices for the zero in the octal; does not lead to any result that every set is compossible: instead zero, as extant every disjunction (even between an essence and itself in any otherwise empty world) is then found present in every K4 subgroup: it is also the last positive property remaining in the partition by eliminiation and is the only set in GF(8) possible and compossible with all Ω, I cannot show that zero will also entail every positive property in its scope in Ω (as does a Godlike essence) unless by a modal collapse (every distinct world is then found necessary, from the liberty of God), for zero is principal overall and absorbed by identity into every essence in GF(8). Each set remains disjoint and incompossible unless found trivially so as with the zero alone. (What, after all, is a positive predicate "plus zero", anyway?)

Zero is then also always extant one side, zero itself only extant as a set alone in an otherwise empty world but for the zero freely exiting both sides. It is also the case that the zero only acts as if from the set of virtue, (as it does in the "Manifold Of Virtue" - see the book "Seven Eyes Open", (Fourth Edition)). Then zero, should itself not be confused with the empty essence lemma placed in the middle, for by the axiom of virtue it is always necessary. If the zero is so defined, it is not as the empty essence, but as all Ω also including itself: it is not empty or a self-difference as defined (but taken without the singleton zero itself, the seven elements of the octal also suffice - and without any self-difference - as a valid identity for the octal group).

As the zero is a virtue, it is also a closed set and minimally so. It may not entail any other positive properties and all GF(8) entails it: the result is a principal element not identifiable with all Ω unless by modal collapse. As the zero is also (in the model) a closed ultrafilter of its very self, by all (any) means of aspiration I may never find another (positive property or) closure properly containing it except for all Ω; there is no result found by aspiration that any other positive property is reachable from the closure and "ultrafilter" of the zero itself; it remains disjoint but also compossible: as it is an element of the octal and it is never empty by definition.

Then the zero (if equivalent to the octal whole), only ever has its own closure broken positively by all possible and positive predicates in Ω: as it is singular, there is self-reference from the closure of the octal to its own closure beyond it: Yet the world is always complete in possibility, as something in itself, as "noumena" and so every possibility of divine choice is then a proper subset of the current world: yet the world is never reducing to φ. The Godhead is possibly never "incomplete" if it is "zero-like". (And then algebraically closed?)

So, aspiration to all Ω from zero I say, is positive, and zero only ever reflects the possible liberty of God in virtue, as when zero acts alone, the world of Ω is unchanged. Then there is a collapse of zero to the set of all virtue (or vice-versa) if zero is ever alone in an otherwise empty world but for God: the least would be every world's equivalent, and if a being is ever "zero", God moves him.

Then the zero entails that under the octal schema in positive properties, every disjunction results in only the possible: the "world" itself is always "possibly Ω" (or possibly something in it) which is the world itself, but freely decidable disjunctions as decided by the Godhead always result in a possible world (not Ω itself unless acted on by zero alone) extant, based on what is possible in Ω. The result carries necessarily: God is omnipotent, the zero alone always reflecting one predicate: of divine liberty (a virtue) necessary to deciding every disjunction

I find no result by inspection or aspiration that zero may be empty or any self-difference. Zero, cannot ever have an empty difference. Why? because it is all Ω also with itself by definition of the octal and its schema. It is an atomic singleton; with no proper subsets. An "empty essence" predicate may not be applied to the zero as much as God in Trinity (and closed on all possibility), is not divisible in GF(8) but to the zero its identity element. If the empty essence is forcibly applied to the zero; then zero becomes incompossible with every other Godlike set in Ω which are preserved in every disjunction, as the zero must always be decided against. As this is a contradiction (all is compossible with zero, the zero is always extant every disjunction; all else as such is an empty world if zero is empty), God exists in His every Godlike essence unchanged, and zero is recovered from GF(8) without the possibility of the world changing by it (for an empty essence is also possibly an essence for the world in Ω, unless it can be shown that the zero is in modal collapse, for as an essence it surely would be so in an otherwise empty world yet present with God.).

Aside:
Then the Revelation of Jesus Christ states the circuit of the least precedes the Judgement on the last day (see the book "Seven Eyes Open", (Fourth Edition)). Therefore, with that possible modal collapse as above, the "least" proves he always exits with his God in every disjunction in a full octal as if found in that circuit.

So, a working objection may begin to look something like the following.

I state that the empty essence lemma may possibly be stated as the middle Ω \ φ ∧ φ of a disjunction:

So φ(y) can be stated as (Ω ∧ Ω \ φ) ∨ φ. As ∧ is an associative relation, and the relation here between possible worlds (compossible sets in the partition of Ω) transitive or possibly only symmetric, I can write the middle as Ω ∧ (Ω \ φ ∧ φ) or equivalent to the application of the empty essence lemma, so that I hold Ω ∧ φ(x).

Then, I find in the disjunction of (Ω ∧ Ω \ φ) ∨ φ the equivalence to (Ω ∨ φ) ∧ (Ω ∨ φ) by De Morgans law, leaving me only with the simple disjunction Ω ∨ φ, extant the left side only.

But then, if P(Ω ∧ Ω \ φ) => P(N(Ω ∧ Ω \ φ)) and that P(N(φ(y)))=>∃x:NE(x) as by the definition of NE, then if φ(y) holds in the middle x∧y, then as x ∨ φ(y) holds in the model and x is non-empty (there is no empty world), N(φ(y)) => N(∃(x)) =>N(¬φ(x)) => "N(⊤)" on that side of the disjunction within the disjoint and separate ultrafilter in all "x" with its own principal element of NE (as if always another individual). That results in the empty essence only ever applying to one side as predicated: that side which is never extant. If the otherwise empty middle x ∧ y is ever found it is negated to inaction, the necessarily complete middle (with an essence) of that predicate φ is always exemplified and positive for, colloquially, Pos(x)<=>¬Pos(¬x) and all of Ω on the left side of Ω ∨ φ is positive and exemplified into action instead. Then, any non-trivial middle is simply found to be "not positive", and is not contradictory or ever "false". The middle may exist, but by God it is never exemplified in GF(8).

The result that the octal's elements present (e_p, e_r, e_s etc.) are only exeunt one side simply makes the predicate φ hold over the middle as if also that one side extant: yet it is only ever found to be empty as a consequence of that free decidability of positive predicates, for any non-trivial middle is not found a predicate ever represented in or of the essence(s) of the ultrafilters, the predicate(s) made of NE. It is true the schema of the octal in the sets of the disjunction leaves the middle always "negative": the octal simply leaves the resulting middle completely empty or positively "inactive", and the singletons in GF(8) themselves are always found to be positive.

Then if the empty essence holds, it does not appear to do so in the middle (as empty both sides, the self difference cannot equal itself as empty without contradiction, all empty "worlds" are identical.) but must be predicated to the side never extant (as God has already "decided" the disjunction and the world has moved with Him), that side then as impossible as ⊥.

A more formal objection made:

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. Under Scott's axioms, the empty essence certainly holds predicated on the 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; 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 above. 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.
3. N.E. (as by Gödel’s definition) states all essences 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 empty 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 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.
5. 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 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) also fulfills the same definition of N.E. with the empty essence, by exiting one side only in a minimal disjunction formed by axiom of virtue.
6. 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.
7. So, a model exists in GF(8) because Gödel’s axioms are found consistent, but they merely lack the axiom of virtue.
8. Dana Scott’s axioms lead to modal collapse but not inconsistency by application of the axiom of virtue.
9. 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.
10. 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 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 aspiration, as by Zorn's lemma) and so no negative (incompossible and also "negative" as necessarily disjoint sets found conjoined (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 (by 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.
11. By aspiration only in the non-negative, Gödel's definition of "positive property" and also "Godlike" therefore permits disjoint and incompossible ultrafilters.
12. So, I 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.)
13. 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.
14. I must also state the absence of any "divine dipole" shared in 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.
15. So, I conclude that Gödel’s axioms are completely valid, requiring but one further addition to his ORIGINAL axioms! (His assumptions are invalid, the axioms are valid.)
16. 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 composed 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, thereby breaking the binding of virtue without also entailing a negative result, and then is not a positive property. Ultrafilters are closed.) The individual x in the essence G(x) is a-priori composite to begin: the definitions and axioms remain valid.
17. So, what is a positive property? Simply that equivalent or set which will entail its very own N.E. 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.
18. The empty essence is then applicable to every individual as every 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.
19. 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. 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 (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).

Summary:

If Scott's axioms are to hold, then the side not extant in every disjunction has an empty essence; the true esence for which has adjoined itself to virtue on the other side chosen extant, to produce the principal element of the result of that binding. (By axiom of virtue).

If Gödel's axioms hold, then the empty essence may also apply across any disjunction in virtue to an incompossible set or ultrafilter. That set is an ultrafilter closed with its own essence of N.E., and the empty essence is then clearly free to apply across the disjunction if it is then assumed that only the principal element was present the side not extant: for then that essence certainly is as N.E., despite there (possibly) being no modal collapse (from Scott's conjunct). If there is a non-trivial set on that side not extant then it is incompossible with that on the other by the above: The empty essence then is the statement that if the disjunction is formed as between the individual x and y, then y, if empty of essence, is equal to the individual x, in that the essence of y entails the N.E. of x. Then these two ultrafilters, with the middle of y=x, entails no inconsistency, for the empty essence across the disjunction is not extant, only the essence of the separate individual x which is necessarily extant (as if already chosen extant) with a predicate of N.E., a variable then bound by the essence of y. (For the essence of any such x remains principal.)

If the essence of x in kind (which is then empty) would entail the N.E. of y (when y is extant), then again, the principal element otherwise in x has adjoined to that of y, which as an essence is clearly equivalent to N.E.

The model of GF(8) does fit (there is no empty essence both sides in any disjunction), the axiom of virtue then requires that the empty essence (if present) not be compossible with any other ultrafilter in the octal. As the middle as above carries the statement that x=y across the disjunction, equal to the empty essence itself, it is not a compossible essential property as there is then an inconsistency. There is no empty ultrafilter; virtuous choice being already extant, there is no empty world to be found.

Besides Gödel's axioms permitting a partition of Ω into separate incompossible ultrafilters, there is little else to claim. I only restate that Gödel's definition of a positive predicate and of the Godlike predicate permit this:

And by the model of GF(8) it is not possible for the empty essence φ to be found both sides of a disjunction (giving an empty world). I then state that it is incompossible. An empty essence simply cannot hold both sides and remain empty of essence.

So, the empty essence φ cannot be the equivalent of any Godlike individual in the partition of Ω (if it is indeed partitioned with GF(8)). As no other predicates are possible for such a Godlike being, the self-difference contains no essence or property in its scope. Then the portion of Ω remaining in the empty self-difference is also empty; it is no essence of that Godlike individual. It may not appear in any freely decidable disjunction with a Godlike essence; only similarly Godlike sets are so freely decided.

As the only possible sets that are compossible with the Godlike are also Godlike, assuming the empty self-difference is also an essence for the Godlike individual in GF(8) leads to a useful contradiction rather than simple antinomy overall, as there can be no empty self difference, only the empty predicate equalling itself, as also excluded from every ultrafilter by axiom. Then as the empty self difference leads to that simple contradiction of equality; it is not an essence for God, it must be completely empty; then the axiom of virtue holds firm every axiom of Gödel's own; with that proposition or lemma negated for the Godlike partitioning Ω. The empty essence is an essence, but not for the Godlike in GF(8).

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