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Ultrafilters Of Trinity

Here again is the definition of an ultrafilter U upon an indexing set I;

1) For any subset S of the indexing set I, either S or the complement of S, (I-S) belongs in the ultrafilter U, but never both.

2) The empty set φ is never a member of the ultrafilter U

3) If sets X and Y are in U, then so is their intersection.

4) If X is a set in U, then so is every set that is a superset of X, (All sets that properly contain X as a subset)

These sets S, X, Y etc... are subsets of the indexing set I, and are not to be confused with the set upon which is the index.


Certainly, if we restrict ourselves to Klein four groups only: Then if The octal itself as our zero element is a member of U, then the empty set is not. Singleton elements correspond to four-groups of Klein four subgroups. If we make our indexing set correspond to just these klein fourgroups, then if X and Y are klein four subgroups, so is their singleton intersection (by correspondence). Their complements are not klein four groups and are not in the ultrafilter. Every Klein four group is a member of U, though we have a division here into two sets. one of singletons corresponding to groups of subgroups, (of which their intersection is the singleton) and the other klein four groups of singletons, the elements of those groups.

We then have a conundrum: are these two sets seperate? or ar they two members of the ultrafilter? For now, we will refer to them as left handed and right handed. Those of singletons will be referred to as left handed, and those of four-groups as right handed. Thus {a,b,c} is left handed and {a} or {{a,b,c},{a,d,e},{a,f,g}} is right handed.

What we have here is a case of an ultrafilter that is certainly non-principal. It is not generated by a single element. The left handed set easily corresponds to having the octal as its additive identity, whereas the right handed set corresponds to having the Klein fourgroup, (in whatever corresponding fashion there is): as its additive identity.

We may easily transform {a,b,c} through the values {{a,b,c},{a,d,e},{a,f,g}} using a seven cycle and the appropriate Frobenius map: so we appear to have an ultrafilter that is happily generated by three elements at least. (As long as we restrict our definition of "set" to subgroup, including the octal as our additive identity.)

Some scripture might help as a guide, it often does!

-- Click To Expand/Collapse Bible Verses -- John ch1:v1-18
Joh 1:1 In the beginning was the Word, and the Word was with God, and the Word was God.
Joh 1:2 The same was in the beginning with God.
Joh 1:3 All things were made by him; and without him was not any thing made that was made.
Joh 1:4 In him was life; and the life was the light of men.
Joh 1:5 And the light shineth in darkness; and the darkness comprehended it not.
Joh 1:6 There was a man sent from God, whose name was John.
Joh 1:7 The same came for a witness, to bear witness of the Light, that all men through him might believe.
Joh 1:8 He was not that Light, but was sent to bear witness of that Light.
Joh 1:9 That was the true Light, which lighteth every man that cometh into the world.
Joh 1:10 He was in the world, and the world was made by him, and the world knew him not.
Joh 1:11 He came unto his own, and his own received him not.
Joh 1:12 But as many as received him, to them gave he power to become the sons of God, even to them that believe on his name:
Joh 1:13 Which were born, not of blood, nor of the will of the flesh, nor of the will of man, but of God.
Joh 1:14 And the Word was made flesh, and dwelt among us, ( and we beheld his glory, the glory as of the only begotten of the Father,) full of grace and truth.
Joh 1:15 John bare witness of him, and cried, saying, This was he of whom I spake, He that cometh after me is preferred before me: for he was before me.
Joh 1:16 And of his fullness have all we received, and grace for grace.
Joh 1:17 For the law was given by Moses, but grace and truth came by Jesus Christ.
Joh 1:18 No man hath seen God at any time; the only begotten Son, which is in the bosom of the Father, he hath declared him.

Christ came to declare the Father to us: Obediently he took on form as a man and, not thinking it robbery to be equal to God, as He is God Himself, he humbled himself and in the perfect display to us of complete and full human faith and obedience, suffered the shame and igmony of the cross, scorning it as nothing. He was resurrected and showed us all the true way and the life. Being God himself, he became the person upon whom we believe, being God revealed to us in person and character: We need no other name under heaven by which to be saved.

So, if God was truly as a man, how can we relate human faith to this trinity model? Christ Himself said to us , "...he that hath seen me hath seen the Father."

There must be some correspondence between the facets of our human experience and the members of the ultrafilter, if we are truly made in the image of God: because God is spirit, and although our form may be his likeness, we would expect to find the limits of our experience somehow within Him, rather than just the shape of our bodies. I do not want to take a trip down gnosis street: more likely I wish to examine some of the concepts with which we associate our faith.

i)The Temporal
We assume God is present with us and isn't restricted to a static state that passes by and is never experienced again

ii)The Spatial
We assume that God is with us in our surroundings, and aware of actual events

iii)The Reconcilable
We assume that human understanding is somehow a subset of God's own experience: and he is not totally alien to our situation.

iv)The Inextensible
We assume that in whatever way we ourselves can imagine the three above, God's own experience and understanding is such that it can not be magnified except upon itself to itself. Or alternatively, it may not be true that a certain set of experiences could be considered infinite, but certainly could not be so greater than their maxima. It is this maximal bound we would wish to find all satisfaction of our faith within.

These concepts are discussed in the next few pages.


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