Self-Reference and Closure

If the octal in completeness self-references one locally closed K4 group, So that say, the sets u, v and u&v are in appearance empty (the set r&s not empty though), then technically the octal's closure is "incomplete" or else I have to account for the properties in r&s.

One solution is as before, to set r = r1 and u = r1-1.

Another solution is to extend the octal over the local closure so there are always some non-empty u, v etc.

When it is noted that the octal "exists" as seven disjoint closed sets, then q becomes a consequence of the partition of the set of all positive properties into p, r, s, u, v etc, and q is not the start of things, or the "jumping off point", but rather the will of omnipotent God as to partitioning the octal itself will suffice to form any required q, as fitting a consequence of that will for any freely decidable act.

I.e. partition the set of all positive properties into the seven disjoint closed subsets as described and then form q = r&u-1 from those seven etc. Then the local closure of {p, r, s} will agree with the case we assumed incomplete "completely".

Then God is not "reactive" but is "genesis" instead.

In every case then, there is a local closure corresponding to a K4 group which is a proper subset of an extended octal in apparent "self-reference" but this is a symptom of the short-sightedness of the imperfect! It may be understood as any one of the above, the latter accounting for both the previous two solutions; there is no sense of failure from God existent. Rather, if the K4 group is finite in positive properties, that does not preclude an infinite set of all positive properties in God as the octal perfected.

So, there is technically no self-reference other than that described in the book in chapter 4 under Godel's incompleteness. The octal's closure is enough for us to find fault with the idea of God in completeness: for in our myopia many things will appear inconsistent; but the set of perfections is still consistent. (We are not able to design our own Gods in honesty: we are required to have faith.)

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