Metaphysics

The Problem Of Zero
By the same argument as on a previous page, if there is an element compossible with every Godlike essence, it must itself be closed without any possibility of further aspiration. An obvious possibility is Ω itself, yet there is also the possibility of a zero, which by the previous page, if possible, must be an atomic predicate and also selfidentical. It, then, must also be present in every disjunction as always extant; and if in modal collapse must correspond to all Ω
I should also note that just as virtue is recovered from the remaining elements of the octal (or C7) in product together whole; so is the zero with the other seven sets in the octal. If the empty essence applies to the zero; it most certainly cannot be composed with the remaining sets which are incompossible as by the previous page and the one before without equalling itself: it is surely empty equalling only itself. As zero exits every disjunction with that essence principal; if it exits the other side, astonishingly (see the book "Seven Eyes Open" (Fourth Edition)), there is then modal collapse and God will always exit that side with His hand; the angel is never found so fallen; and remains a "morning star".
The problem occurs with zero as compossible with all seven sets; he is present alone never: only within each disjunction as with whichsoever is that set extant, the free choice being also his as a valid essence. (Either God moves him or with him.) He always exits the disjunction with God and His essence(s). The disjunction, then, is always decided; and then one set at most attains the empty essence.
The zero never appears in a decided disjunction alone without modal collapse and with the other side not empty of essence. God moves with the zero always. Zero is always absorbed into a Godlike essence; the compossibility is ever one of identity, the sets remain disjoint; else there must be modal collapse: some sets in the octal become otherwise empty but (possibly) for their essence(s).
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 as the set of all Ω) with the empty selfdifference by placing that operation upon the sets of positive properties alone; the result that the whole octal indeed 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 in the model remaining in the partition by elimination 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 would then be 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 present in a disjunction alone with some other set. (What, after all, is a positive predicate "plus zero", anyway? God entering His creation is a far better prospect.)
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 possibly as all Ω also then including itself: it is not empty or a selfdifference as defined (but taken without the singleton zero itself, the seven elements of the octal also suffice  and without any selfdifference  as a valid identity for the octal group and over positive properties).
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 selfreference 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 "zerolike". (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 equal the set of all virtue (or viceversa) 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" and virtue, God moves him; and moves him as the word of God.
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: that 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 selfdifference. 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 each of 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 itself 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 for zero being 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. The least makes the simple choice to overcome. (I.e. to aspire to a complete God according to all proper faith grounded on only sound doctrine, that aspiration then found complete in a "new name".)
As to Revelation and (in)compossible essences (the seven churches?), if the zero is in modal collapse in circuit (as in the "mystery of the seven golden candlesticks"), then there is possibly an eightcycle over the octal from his circuit, as across sets, x = y only with the completion of that circuit, one spanning all Ω, as made in completed octals that are certainly never disjoint in the "new name"; there is a union made in octal "addition", rather than a sense of x = y modulo 8. (Although that is the case too if one realises it!) So, as those octals are each a model upon Ω and a partition, when the operation is made in that "crown" of union instead, every set becomes as a virtue bound as to otherwise incompossible sets (a "union" is the binding function then chosen, i.e. the "marriage of the lamb"). God is so proven to be able to enter into His creation with zero in modal collapse; nothing positive at all is ever spared instance (my own great hope, as for a paradise). Then begins the reign of "omnipotent God" (cf. Rev 19:6). God is then able to enter creation with just such a union leading the way. (I.e. as to the "second coming", as proven perfectly that as it occurs, He has then entered in without either He, or the world, ever being diminished or privated.)
Union, then, an operation acting as if virtue bound is certainly closed by the axiom of virtue; entailing neither set/model present in disjunction but for itself as virtue closed on their union. It cannot aspire further than Ω, as is each operand is equal to its own result, the difference is only the binding (a function) made with union due to the other bindings with union made between the other seven sets. There is an eightcycle if that circuit is found necessary; only the distributive laws are then required; for that, the actual multiplication in C8 over the octal may be done only on elements which are "equal" as zeroes and possibly also with "unities" (Ω and as virtue) otherwise there is no solution. The "new name" instead, requires a identity of 1 = 0. So, unless every element is also as the like (1 = 0) placed under an eight cycle, there may never be any such sense of the distributive laws. Ω, similar to zero in modal collapse is ever a null ring; and the union of an eightcycle over those octals can only be a symmetry of algebraic closure present only at that very limit.
Finding such symmetry is not easy  an analogy could be the factoring of zero into its possible divisors.
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