Holy Trinity - The Only Finite Godhead

Since the number of subgroups rapidly diverge from equalling each other in number, from one extreme of an additive subgroup of a finite field - trivial subgroup to entire Field, there quickly become not enough elements in a Field's multiplicative group to cycle through every subgroup... We find we need more than one such multiplicative group. It follows, that with other primes and other degrees of extension our system is unbounded and diverges to infinity. There is in every other case an ever present lack of correspondence, that beyond a factor of one.

Since all others diverge, the only finite description of the Godhead's symmetries is the system of the Galois fields GF(2), GF(4) and GF(8). Thus, though these are different, they possess a singular description unique to the concept of a Holy Trinity.

However... Could it be true that there are many such Holy Trinities, all with the same structure? Or perhaps is there a whole plethora of "gods" ANY three of which when together satisfy the structural symmetry of the Holy Trinity?

Paul's sermon to the Greeks at the aereopagus in the biblical Book of Acts (ch. 17 v. 28), defined God in the manner "For in him we live, and move, and have our being; as certain also of your own poets have said, For we are also his offspring."

God has perfect knowledge of the state of things by argument, and is well aware of other "gods" - for to be perfect, such others must live and move and have their being in HIM. God is in no doubt as to in whom he has Glory. Such perfection of Godhood belongs in such circumstance to but the three members of the Holy Trinity. Therefore these three are unique in instance. In the bible, God himself declared he is a jealous God.

Psa 148:13 Let them praise the name of the LORD: for his name alone is excellent; his glory is above the earth and heaven.

Isa 42:8 I am the LORD: that is my name: and my glory will I not give to another, neither my praise to graven images.

-- Click To Expand/Collapse Bible Verses -- Isa ch43:v9-13
Isa 43:9 Let all the nations be gathered together, and let the people be assembled: who among them can declare this, and show us former things? let them bring forth their witnesses, that they may be justified: or let them hear, and say, It is truth.
Isa 43:10 Ye are my witnesses, saith the LORD, and my servant whom I have chosen: that ye may know and believe me, and understand that I am he: before me there was no God formed, neither shall there be after me.
Isa 43:11 I, even I, am the LORD; and beside me there is no savior.
Isa 43:12 I have declared, and have saved, and I have showed, when there was no strange god among you: therefore ye are my witnesses, saith the LORD, that I am God.
Isa 43:13 Yea, before the day was I am he; and there is none that can deliver out of my hand: I will work, and who shall let it?

UPDATE - I don't want to leave you thinking that the maths is pointless, or it can not be understood... Whilst I at the same time am a little hazy on precisely what "operation" belongs to the model of K4, in declaring precisely and rigorously what is meant by O8 divided by C7, (O8 the octal and C7 the cyclic group of order seven). Or indeed any other variation on elements, their inverses and the "+" and "*". Mathematicians are happy considering the Galois field of order 8 as a single object albeit with two binary operations, yet have decided upon naming the elements as primitive roots of unity (A competent solution for the multiplicative part) whilst not providing an explanation for how these elements equate to the additive operations.

If you are to walk away knowing anything from the modelling math, the one thing within which the most benefit is found are the reasons for the model itself. That God, being perfect requires a Trinity in which He may perceive Himself as existent in order to know that His existence, (at least from His own point of view) is better (more perfect) when considered outside of His Self than within it. Else God may not know that He is perfect in the same way as we understand it from Anselm's argument. In fact, the Trinity is required by Anselm if God is perfect and vice-versa. Arguing from a similar basis as would Hartshorne, that Perfect God not only requires Perfect knowledge and could not conceive of the possibility of His own non-existence, but that Perfect knowledge would include any fact we could "know to be true". In this case His existence by the method of Anselm.

God requires to know His existence through another, and far from implying the idiocy that to kill all believers kills God; that He is somehow fed through a crop of faith, These three must have existed from the beginning as logic dictates even from a "proto-Anselm."

Whilst each knows of their own self, we see the non zero K4 elements as one whole see themselves extended into the Octal on the one hand and distributed through the octal on the other, thus 'glorified'... whilst the Octal witness themselves as one element of K4 as well as a "deeper, more recurrent" distributive lattice where single elements of the Octal correspond to K4 groups... Finally the C7 elements retrieve the same treatment of the Octal, but more keenly "On the other hand". There is extension of self through the other elements apart from the identity map from each to their own.

It is the extended properties of each of the elements that allow God to see Himself not by only Himself, but as God would wish (and requires) to be seen, (within the only possible vessels that can fully glorify God) and to all intents and purposes justify His existence and provide Him with the "Am" part of Descartes' most famous predicate.

Could it all be that simple, "I Am that I Am"? As a Trinity and a complete one at that, the name itself suffices for all three.

If there is some concern within you that the necessity for a third member of the Holy Trinity has not been adequately treated, then I give it another treatment Here. Look for the section labelled "UPDATE" for just that. Bookmark link here.

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