L(G) Generates An Ultrafilter
We begin by defining an ultrafilter. An ultrafilter is a set of subsets of a set that result in a clear division, either a subset is in the ultrafilter or not. The sets corresponds to the following proeprties which are chosen so that the sets in the ultrafilter maintain some form of "largeness" as subsets in the collection.
We define an ultrafilter U as a set of subsets of a set so that:
1) The empty set does not belong to U (it is "small" we do not want it, as it screws up the following.)
2) If sets X and Y are in U then so is their intersection. (they maintain some form of "largeness".)
3) If a set X in U is contained in a set Y
the Y must be in U also. ("bigger" sets than X with the same elements are also members.)
4)
For any subset X of the collection M, either X is in U or M\X is in U (a set is large, or it's complement is large.)
So we can state that if an ultrafilter is the collection of all sets in M containing some single element 'm' from M, the above is satisfied. This type of filter is called a "prncipal" ultrafilter with the generator 'm' called the principal element.
So, how do we actually show L(G) entails all these properties?
L(G) is a virtue: and since it implies that God may choose any set of virtues He wishes, we may assume that there is no virtue, or choice of virtue or any other positive property of God that can be privated by L(G). Likewise unless one is a member of the Holy Trinity L(G) is universally correct. God is free to set virtues and commandments for men.
We first define some notation for the sets of beliefs that an individual may have of God. Before we used an equivalence relation '~' however we need some sense of commonality between individuals, so we state.
H_{x}(y) is the set of beliefs of God held by the individual x also held by the individual y.
Then H_{x}(x) is simply the set of beliefs held of God by 'x'.
Likewise H_{G}(G) is the perfect set of beliefs held of God Himself (Jesus) Of course we could use L(H) also but for now we will use L(G). As Jesus is God, He automatically holds L(G).
We will however deal more rigourously, if H_{x}(x) contains L(G) then if there were a fault 'f' on x before God, then L(G) requires of the believer that f be repented of in order for God to redeem x. L(G) is the element that generates the sovereignty of God over the man. If we simply state that for every f(x) at time t1 there is *always* a time t2 afterwards at which f is repented of by x and thence f&¬f holds and f can not apply. L(G) is the property held by x that ensures this.
We will only assume that H_{x}(x) contains positive properties held by x of God, and that some of these may private the virtues held in H_{G}(G). If that is the case, we state that x is at a fault 'f'.
if x holds every positive property except for L(G) then x may also assume some freedom to induce a shift or transformation upon the virtues (fixed in Christ) to a set of virtues where some positive property 'p' is attached and some 'q' dropped out. Such freedom is a fault when not done perfectly. Thus, we may assume that since the authority or perfection is not found outside of the Holy Trinity to do this, (Jesus Christ, the same yesterday, today and forever) then the set H_{G}(G) \ L(G) is also equivalent to a fault 'f'. One would presume to have the judgement!
*H*_{G}(G) \ L(G) would generate a G v p statement also: for if an individual purposed a transformation on virtues and asked for a truly positive property in prayer that he perceived was not present, he would incur a fault with another virtue dropping out (a clear fault then!) (**) He would hold L(X) instead of L(G).
Likewise, For a subset of virtues held by x, H_{G}(x) \ L(G) one would have immediately made ones own law to ignore the rest.(***) virtue would have been privated; not by contradiction as per the above but by exclusion.
So we may simply check off the conditions for an ultrafilter
1) This holds simply because L(G) is present: It entails that every commandment of God is made to ensure the persistence of one unique set of virtues.*
2) The intersection of two subsets of the same law are also a subset of the same law when both contain L(G). (Faith then, is common on virtue found in obeying the one law. See (*) above)
3) L(G) ensures deferrence is given to the whole of the law - supersets of virtues are positive: (Up to virtue becoming over constrained) - see (**) above. Also, a subset of virtue is not virtuous without L(G) - one would make ones own law! See (***) and (4) next.
4) virtue is privated or not.
One might ask why the distinction between (**) and (***)? Clearly there must be a universal law against (**), so that every subset would be properly defined? Then we are justified by faith: for not holding L(G) would be to have some other faith L(X) with another law.
Some Identities
- i) {H
_{x}(x) intersect H_{y}(y)} = H_{x}(y) = H_{y}(x)
- ii) H
_{G}(x) = H_{x}(x) if x holds L(G)
- iii) H
_{G}(G) always *strictly* contains H_{G}(x) = H_{x}(x) if and only if 'x' (properly not equal to G) holds L(G) (see (**) and (***) above)
- iv) {H
_{G}(x) ∩ H_{G}(y)} equals H_{x}(y) if both x, y hold L(G)
So, (1) to (4) the conditions for an ultrafilter hold with,
- (1) holds by definition, includes L(G)
- (2) holds with (i) (ii), (iii) and (iv),
- (3) holds with (iii)
- (4) holds with (ii) and (iii).
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