Existence Again
Now given r&u^{-1} = r1 say, (u=r1^{-1} etc) I once more may revisit the subject that "Existence and rest are equivalent" in chapter 4 of the book.
Now, clearly given ΩvL(G)&Ω^{-1}=> e, I have on rearranging (with <e> the set Ω is made principal with "e" as the generative element), <e> v ∅
Then I have the same result with u^{-1} v L(G)&u=>
v^{-1} by symmetry. Other results within the octal follow; but those results with (r&s)^{-1}, u&v etc become simply as the disjunction ∅=(Ω)^{-1} v L(G)&(Ω)=>(Ω)&e=><e> or reversed as (Ω)vL(G)&(Ω)^{-1}=>e.
In fact, (r&s)^{-1}v L(G)&(r&s)=>(u&v) is then rephrased as the disjunction ∅=(<e>)^{-1} v L(G)&(Ω)&e=><e>. (The right hand side becomes necessity.) Truly, there are some strange effects! (I reservedly posit that there are always some separate, disjoint and distinct sets of properties u, v etc in the completed set indexed by the octal ultrafilter. We do not choose Ω "looking up" God instead chooses "looking down".)
So, there are some disjunctions that are not freely decidable, those in ∅=(Ω)^{-1} v L(G)&(Ω)=>(Ω)&e=><e> have the left hand side negative and the right hand side necessary. (Clearly God can not rest from existing for long (if at all) being necessarily existent); It makes sense to state that in like terms (with "r maximised to all q"), the rest of the octal is found following suit, with u=r^{-1} etc, so that there are no "empty sets", it then becomes necessary that some sets become "empty" in the octal, but these disjunctions have been deliberately maximised!
Symmetrically those disjunctions rearrange to simply "rest" upon existence, so that Ω v L(G)&e, which is not illegal at all: and from Chapter 2, in the section "Liberty is Free to Rest", "e", a set which is not negative, is always closed and entails no other property - it may be adjoined to either side of the disjunction (or both sides) without much in the way of confusion.
So, ∅=(Ω)^{-1} v L(G)&(Ω)=>(Ω)&e=><e> is become e v L(G)&(Ω)=>(Ω)&e=><e>. That is to state, N¬(L(G)&Ω&¬<e>) or indeed its equivalent N¬(L(G)&(Ω)^{-1}&¬e).
That is, when Ω is stated not to be "god-like" (with existence as a principal element).
Then, in the octal there is no "empty set", as <e>^{c}=∅. and the ultrafilter is in this case indexing the set of all positive properties <e>.
Then the sets <e> and "e" alone may be of the same disjunction rearranged, but God may rest showing either nothing, or rest as existent in all things perfected. Existence as a predicate itself is recovered from every act of liberty, or as necessary existence in the ultrafilter of Godel's proof (Anselms filter made principal as <e>, "r" maximised to become as all "q").
Now, "e" is found on the side of the disjunction on which liberty is found to act. When "r" is become maximised, "r" also fails to become necessary as it yet lacks "e", (existence, rest). such r=q is enough to show logical consistency of all positive properties but it is entirely an act of symmetry to observe (by rearranging) the same disjunction to imply either God's necessary existence in all perfection as <e> without logical alternative or indeed, simply rearranged as existence entailing nothing, logically opposed to that same consistency that entails no existence unless one would also predicate necessary existence as to construct either Ω&e or reason as Godel in constructing <e>.
So, e v Ω does not or may not convince anyone that necessary existence entails all perfection (or indeed any set of positive properties, e being completely closed without consequence); and <e> v ∅ also appears to entail complete modal collapse; were it not for the fact that this is the same disjunction rearranged! (Or indeed, the set of all positive properties is somewhat limited to a proper subset, - one set of seven in the set of all positive properties, forming with "e" the octal group.)
There is another possibility - but not supplied with any proof:
If any positive property, even Ω is only found local to a K4 group of the octal, and the positive properties of the octal are always extended beyond any particular choice of Ω, so that there are always found another set of six subsets (and "e") to complete a full octal, then God is extensible by self-reference (possibly): and even if there are found sets of <e> and ∅ there will instead be no such empty sets; every positive property now found, merely generates one set of seven DISJOINT sets, rather than all the octal ultrafilter in just one set: For then, the closure of all positive properties will always outrun every K4 group closure (those being generated by any "singleton" positive property), even if it generates whole the ultrafilter as we would in naiveity conceive of it.
I much prefer this idea, as God is free to choose for Himself any such disjunction with "r" maximised (to q), with no disjunction ever found to entail any empty set: for those disjunctions are then always part of a domain - an extended system over the local closure: for any closure of positive properties is found to be simply a local closure of a K4 group, and never "God" perfected in the octal.
So, when Descartes was reasoning that only God was "infinite" but mere ordinal quantifiers "indeterminate" perhaps he was on to something, though we now know that there are many such infinities. God it would appear, is free to move His own goalposts!
Were a coherent "set of everything" (God) able to be so devised, it may follow after such a fashion as to be indexed by an octal in self reference as above.
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