Liberty Generates The Ultrafilter
The construction of the ultrafilter was such as to state that every positive property following from an application of virtue p&q^{-1} is generated by a principal element, the liberty L(G) due to sovereign God. In fact the ultrafilter was constructed along the lines that every set in the ultrafilter was constructed on the basis as to whether it was positively exemplifiable - that is, could it be performed in a positive manner, and if not then it must be positive for it not to be so.
For every positive property in the ultrafilter s_{i} and every positive property v_{j}^{-1} (v not positively exemplifiable) I have an ultrafilter that acts not only on positive properties but including all properties not positive whose negations are positive. It makes complete sense that God may bring about "evil" to accomplish a good.
Then in metaphysical terms, I may understand that some evil, say, r&s is a necessary consequence of work already done in the indexing set {p, r, s^{}} etc. (Evil superposes on every good work.)
The sets in the ultrafilter of p&q^{-1} => s&v^{-1} give us a series of conjunctive terms, where liberty is always present as a "p". Liberty can alter the ultrafilter so that s=> s_{i} for all or any s_{i} in the filter. So, given that there is the liberty to rest on q^{-1}, I must have q^{-1} at rest positive, and there is always a e_{q}, rather than "v_{j}^{-1}". Similarly, L(G) is always a "p", and p may rest and imply whatever follows as s, an equivalence to virtue.
But the intersection of all the sets in s&v^{-1} is also in the filter, so at the very least there is liberty and rest on q. Liberty generates the ultrafilter. q^{-1} alone does not generate every positive property; but if L(G) may select any applicable p, then I have any (consistent) consequent s&v^{-1}.
Even if there is no sense of complete rest on q^{-1}, there is always one such q^{-1}, being e_{q}. As p&q^{-1} always entails e_{p}&e_{q} it is an observation that q will also contain e_{q}_{} on the other side of the disjunction. What is less clear is that liberty may entail any set of positive properties from the list of s_{i} conjoined in s. Liberty may rest on any s_{i} in s, effectively "moving" an s_{i} to some v_{j}.
Then liberty rests rather than acts. Although I have the converse in u and v in the octal!
As long as there is one s_{i} in s, following from p&q^{-1} the outcome is positive and clearly so: So, as such a set it is in the ultrafilter. Every possible s containing this single term is positive and there appears to be a non-principal ultrafilter with every possible single term "s". Yet the virtue p itself is also consequent, and a positive property (and liberty always positive): I should however state what liberty actually is!
Liberty is a virtue; then it becomes negative if it is required to act out of anything but freedom. Liberty is not "forced" and acts freely only. So, liberty is free to rest rather than required to act positively.
So, s was "in the filter" and v "not in the filter". Then the positive property q leaves p&q^{-1} to entail all in the filter and then p most certainly and necessarily so. As the only single p always present is liberty, it generates the Ultrafilter and there is no non-principal ultrafilter in virtue, as every superset of liberty contains sets with at least one more member, and there is certainly always the intersection of liberty present.
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