What qualifies the 3 entities of the Trinity as one?

The trinity is not something that Humans can imitate, emulate, or fully grasp. God has chosen to express himself in such a way. Isaiah 55:8-9

“For My thoughts are not your thoughts, Nor are your ways My ways,” says the Lord. 9 “For as the heavens are higher than the earth, So are My ways higher than your ways, And My thoughts than your thoughts.

Ill give you my favorite analogy for the Trinity. Note that this is not a "perfect" analogy. There is no perfect analogy. It will fall short in some ways. but there is also truth to it to help you understand.

First off, here is one analogy commonly used that i Dislike People say, the trinity is like water, one water, found in 3 different states (Ice, Vapor, and Liquid).

But this is actually a dangerous analogy, because it kind of applies whats called Modalism. Modalism is the idea that God expresses himself in any one of three forms at any give time, like an actor putting on different characters. This is not truth of the trinity. (In scripture many times we see multiple memebers of the trinity at the same time) One common example is the baptism of Jesus in Matthew 3:16-17

16 When He had been baptized, Jesus came up immediately from the water; and behold, the heavens were opened to Him, and He[a] saw the Spirit of God descending like a dove and alighting upon Him. 17 And suddenly a voice came from heaven, saying, “This is My beloved Son, in whom I am well pleased.”

Note the Spirit as a dove, the Son as Jesus the Man, and the Father speaking from heaven.

This analogy leads to this because water Either freezes, liquifies, or Melts. you cant really have all three together for any length of time. And the Trinity is the idea that all three Persons are existing together eternally.

So now lets get to another anology that i like a lot better. Its not perfect, but it gets across some fundamental truths.

Imagine the United States Federal Government. The Federal Government is One government.

However it is made up of Three Distinct Branches The Judicial branch(Supreme court), the Executive branch(President), and The Legislative branch (Congress).

Now The Congress is NOT the president. The President is NOT the supreme court.

However the Congress IS the government. the President IS the government. Yet the complete picture of the Federal government includes the fullness of all three branches.

Now the members of the trinity are God and are unlimited so limited human analogies all fall short somewhere. But this one really gets across some basic truths i believe. Hope this helps!

In the Gospel of John, we see the 'Godhead' repeatedly displayed.

• As a disclaimer, I would say first that trying to use a single verse to explain God is both incomplete and dangerous in that it can cause a person to be on a slippery slope--humanizing God.

With that said, John 1 v.1 identifies Jesus as the Word, that He was with God in the beginning, and that He was (is) God. V.3 says He made everything (ref. HEB 11:3), v.10 says He was (now) in the world, and v.14 that the 'Word' was made flesh.

For Jesus to pray to the Father shows He gave up His heavenly place to take on human form to fulfill both God's plan and prophesies spoken of before. V.2 says He was with God (emphasis mine). Genesis says that God created the heavens and the earth; it also clearly says that God 'said' (spoken Word), and it happened.

Genesis also says 'the Spirit of God' moved: again referencing the Truine aspect of God. Again, we can see it in Genesis 2:7, in that we were made in His image: created of something physical (dust of the earth), something spiritual (the breath of life), and something soulish (a living soul). I use 'soulish because a better word escapes me now, and I'm not nearly as educated as many of you probably are. However, God the Father is often associated with the mind, will, and emotions as opposed to spirit or body.

Back to John. John 15:26 shows us Jesus--God as Man--telling that when He returns to the Father He will send the Spirit, which is from the Father. Again, in John 16:7-16, Jesus tells of what He will do and why, placing significant distinction between the three.

These are only examples from a single book, and arguably, a single author, and outside of Genesis, New Testament-only which would not satisfy those who do not believe the Messiah has already been here once. It may also help to make the arguably weak comparison to us made in His Image, albeit, taken from a strictly intellectual position fails to paint a full picture.

To preface, Romans 1:19-22 tells us that (1) that which can be known of God is manifest in (them); (2) God has shown it; (3) the invisible things of God are clearly seen from the creation of the world (whether you interpret this to be 'from the beginning' or 'by the examples of creation' it still shows He made it possible for us to see it); (4) His eternal power and Godhead are revealed to us. Having said this, human beings are body, soul, and spirit (Gen.2:7), and we are made in His Image (Gen.1:26-27).

Time would expire to include every reference in the Old Testament of God's mind, will, and emotions, an integral factor in understanding the Godhead and how it is demonstrated in His ultimate creation, MAN.

Not only does the scripture show that the Son is the Creator, but it also shows the 'physica' aspect in the New Testament by the birth and life of Jesus. The repeated references in scripture to the Spirit of God, separate from the Father, demonstrate that although He is unknowable in His entirety while we are on earth, as per Romans, the Godhead is reflected in His complex associations and operations throughout human history. In a similar fashion that we can imagine something, and then create it, but we are not 'different' entities in so doing, so is God as we are made in His image.

Before someone strikes the Spirit part of this type of analogy, I'll posit the HE is the Creator, we the creation, and it is thus completely comprehensible that we cannot send our spirit out to perform actions--that would make us equal with God. The 'spirit' portion of this type of analogy is supported by His 'making' us a living soul by the breath of life (spirit), thereby completing the truine-aspect of corruptible man made in the image of the Incorruptible God. Here I need to close by reminding myself what it says in I Corinthians 1:25, 27-29, and I Corinthians 13:12...that God's far more knowledgeable and wiser than I, and I only see part, only as much as He deemed necessary in order for me to follow...otherwise, if I knew everything, it wouldn't be faith...

So a few notes on your terminilogy in your question from the perspective of the orthodox, canonical viewpoint. As otherwise noted, you simply will not be able to comprehend the Trinity - we are unable to comprehend the incomprehensible God. In A.W. Tozer's book "Knowledge of the Holy" (pages 7-8) he describes this problem thusly:

The effort of inspired men to express the ineffable has placed a great strain upon both thought and language in the Holy Scriptures. These being often a revelation of a world above nature, and the minds for which they were written being a part of nature, the writers are compelled to use a great many “like” words to make themselves understood.

When the Spirit would acquaint us with something that lies beyond the field of our knowledge, He tells us that this thing is like something we already know, but He is always careful to phrase His description so as to save us from slavish literalism. For example, when the prophet Ezekiel saw heaven opened and beheld visions of God, he found himself looking at that which he had no language to describe. What he was seeing was wholly different from anything he had ever known before, so he fell back upon the language of resemblance. “As for the likeness of the living creatures, their appearance was like burning coals of fire.” The nearer he approaches to the burning throne the less sure his words become: “And above the firmament that was over their heads was the likeness of a throne, as the appearance of a sapphire stone: and upon the likeness of the throne was the likeness as the appearance of a man above upon it. And I saw as the colour of amber, as the appearance of fire round about within it.... This was the appearance of the likeness of the glory of the Lord.”

Strange as this language is, it still does not create the impression of unreality. One gathers that the whole scene is very real but entirely alien to anything men know on earth. So, in order to convey an idea of what he sees, the prophet must employ such words as “likeness,” “appearance,” “as it were,” and “the likeness of the appearance.” Even the throne becomes “the appearance of a throne” and He that sits upon it, though like a man, is so unlike one that He can be described only as “the likeness of the appearance of a man.”

...

When we try to imagine what God is like we must of necessity use that-which-is-not God as the raw material for our minds to work on; hence whatever we visualize God to be, He is not, for we have constructed our image out of that which He has made and what He has made is not God. If we insist upon trying to imagine Him, we end with an idol, made not with hands but with thoughts; and an idol of the mind is as offensive to God as an idol of the hand. ”The intellect knoweth that it is ignorant of Thee,” said Nicholas of Cusa, “because it knoweth Thou canst not be known, unless the unknowable could be known, and the invisible beheld, and the inaccessible attained.”

”If anyone should set forth any concept by which Thou canst be conceived,” says Nicholas again, “I know that that concept is not a concept of Thee, for every concept is ended in the wall of Paradise.... So too, if any were to tell of the understanding of Thee, wishing to supply a means whereby Thou mightest be understood, this man is yet far from Thee.... forasmuch as Thou art absolute above all the concepts which any man can frame.”

Left to ourselves we tend immediately to reduce God to manageable terms. We want to get Him where we can use Him, or at least know where He is when we need Him. We want a God we can in some measure control. We need the feeling of security that comes from knowing what God is like, and what He is like is of course a composite of all the religious pictures we have seen, all the best people we have known or heard about, and all the sublime ideas we have entertained.

So that being said, my notes:

• "the Trinity is not one entity with three different aspects"

  Correct; This is a heresy know as Modalism or Sabellianism

• "But rather three entities with different personalities with a shared essence and relationship with each other"

  Incorrect; this is a heresy known as Arianism which was expressly spoke against at the First Council of Nicea

• "the Son, the Father, and the Holy Ghost wouldn't be ONE God, would they?"

  Yes, they/he would be. This is a paradox. God is both one God and 3 persons (not to be confused with IN 3 persons - as if God could be divided) who shares a single divine will and yet has 3 different and distinct wills. Similarly it is not accurate to say that God is 3 persons or that God is one. It is only accurate to say both.

So, now to your analogies:

While you and your friends may share the attribute of being human this is not the same as being "of the same substance" or ὁμοούσιος (Homooúsios). (in fact, being created in the image of God, we share attributes with God) The same is true in all of your analogies. All of these similar objects are ὁμοιούσιος (Homoioúsios) or of a similar substance. In fact, it may be said that only the members of the Godhead are ὁμοούσιος (Homooúsios) - a new classification that was developed to describe God. It may be helpful to explore the origins of this idea - these ideas had been percolating for some time before Christ arived and these ideas were subjects of discussion by the ancient Greek philosophers.. It may therefore be helpful to read an overview of their discussion so as to contextualize the terminology used that often sounds strange to modern ears and understand why the concept of οὐσία or "substance" was adopted for the purposes of discussion by early Christian thinkers. In your analogy of the Greek gods, they are Distinct from the concept of the Trinity in that each God is clearly seperate from the next - the Triune Yahweh is indivisible and Jesus and the Holy are both seperate from God and yet the boundaries and distinction between them are not. These three repsent one single indistinguishable entity with no clear seperation between the three members of the Godhead.

Finally, to address your opinion that Christians are trying too hard to market this as monotheism, I would encourage you to take this idea seriously and consider it fully. As explained here, while one has salvation upon "believing in" Jesus, very quickly the question of who you actually "believe" Jesus to be very becomes important. If you do not believe who he said he was (eg, his statements about the Trinity, then you do not actually believe in Jesus, but something else.

So in summary,

What is it that makes them out to be one? in which sense are they one?

They are one in the sense that they are ὁμοούσιος (Homooúsios) - something for which the translation "same-substance" is not quite an accurate as we don't quite have a modern concept or term for (outside of "triune" that is).

What qualifies the 3 entities of the Trinity as one?

There are not three entities or beings—three Gods. There are, within, or intrinsic to, the one God (or rather the one God is definitionally in some respect three; namely, He is tri-personal) three 'somethings.' They happen to be revealed to be quite Personal.

Put simply, 'trinity' describes something of the intrinsic nature of God; 'one' describes the numeric total of essences that are called God and have the attributes of God: there being but one, (Deut 6:4) who is known variously by titles YHVH, the God of Abraham, Isaac and Jacob, etc.

So they are one because they are intrinsic to each other and co-definitional (like how, analogically, 'Father' is meaningless without 'Son')—each no less than the other being definitional to the one essence of God: God is eternally three in some sense, yet in that He is but one, only God, He is one and only.

The name or title 'God' can, then, understandably, and readily, be attributed to any of the Persons of the Holy Trinity because each is no less God than the other, since they are God for the same ultimate reason: their essence, nature, being is one and the same: the divine and eternal, ineffable Deity, the only true God.

Scripture happens to use a convention whereby the Father, since He is the ontological beginning of the Godhead (not the temporal—the Son and the Spirit are not created, especially since they are eternal, but rather 'Son' simply 'logically' or 'onotologically' follows 'Father' and so forth: the economy of the Godhead is such that the Son and the Spirit recieve being God from the Father, not that they are created) is used to simply refer to the one God. But since the Son and the Spirit are directly of the same substance of the Father, they are just as much God as He: there was no time when the Father was God without the Son and the Spirit being God.

There is 'one' God because 'one' describes not the number of Persons God has, naturally, but the number of Gods. There is but one God, who is, eternally, tri-personal. And ineffable, so that all excercises in forcing God to conform to the current human understanding of the world etc. is quite futile. In fact, we only know God by analogy alone.