What is the Trinity?
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The word "trinity" is not used in the Bible, but the concept is apparent in the text.  Trinity is used to describe God, Christ, and the Holy Spirit as three-in-one because sometimes the Bible represents them as distinct individuals and other times as one and the same.

Revelation contains a clear example of God and Christ being both the same person and two distinct persons.  God and Jesus both describe themselves as Alpha and Omega.  Alpha is the first letter of the Greek alphabet; omega is the last.  There can only be one first and one last.  If God says He is the Alpha and Omega, and Jesus says He is the Alpha and Omega, then God and Jesus are the same person.

Rev 1:8
8  "I am the Alpha and the Omega," says the Lord God, "who is, and who was, and who is to come, the Almighty."

First this verse says that God is the Alpha and the Omega. Then watch as it switches to Jesus as the Alpha and the Omega.

Rev 2:8,
8  "To the angel of the church in Smyrna write: These are the words of him who is the First and the Last, who died and came to life again.

Rev 21:6
6  He said to me: "It is done. I am the Alpha and the Omega, the Beginning and the End. To him who is thirsty I will give to drink without cost from the spring of the water of life.

Revelation 22:12-18
12  "Behold, I am coming soon! My reward is with me, and I will give to everyone according to what he has done.
13  I am the Alpha and the Omega, the First and the Last, the Beginning and the End.
14  "Blessed are those who wash their robes, that they may have the right to the tree of life and may go through the gates into the city.
15  Outside are the dogs, those who practice magic arts, the sexually immoral, the murderers, the idolaters and everyone who loves and practices falsehood.
16  "I, Jesus, have sent my angel to give you this testimony for the churches. I am the Root and the Offspring of David, and the bright Morning Star."
17 When I saw him, I fell at his feet as though dead. Then he placed his right hand on me and said: "Do not be afraid. I am the First and the Last.
18 I am the Living One; I was dead, and behold I am alive for ever and ever! And I hold the keys of death and Hades.
 

Here are more examples of three separate persons, the Father, the Son, and the Holy Spirit, who sound like the same person:
Luke 12:8-12
8  "I tell you, whoever acknowledges me before men, the Son of Man will also acknowledge him before the angels of God.
9  But he who disowns me before men will be disowned before the angels of God.
10  And everyone who speaks a word against the Son of Man will be forgiven, but anyone who blasphemes against the Holy Spirit will not be forgiven.
11  "When you are brought before synagogues, rulers and authorities, do not worry about how you will defend yourselves or what you will say,
12  for the Holy Spirit will teach you at that time what you should  say."

John 14:16-17
16  And I will ask the Father, and he will give you another Counselor to be with you forever--
17  the Spirit of truth. The world cannot accept him, because it neither  sees him nor knows him. But you know him, for he lives with you  and will be [73] in you.

John 14:26
26  But the Counselor, the Holy Spirit, whom the Father will send in my name, will teach you all things and will remind you of everything I have said to you.

John 15:26,
26  "When the Counselor comes, whom I will send to you from the  Father, the Spirit of truth who goes out from the Father, he will  testify about me.

John 16:13
13  But when he, the Spirit of truth, comes, he will guide you into all  truth. He will not speak on his own; he will speak only what he hears, and he will tell you what is yet to come.

Acts 13:2
2  While they were worshiping the Lord and fasting, the Holy Spirit said, "Set apart for me Barnabas and Saul for the work to which I have called them."

John 1:1-2
1  In the beginning was the Word, and the Word was with God, and the Word was God.
2  He was with God in the beginning.

Eze 36:27
27  And I will put my Spirit in you and move you to follow my decrees and be careful to keep my laws.
28  You will live in the land I gave your forefathers; you will be my people, and I will be your God.

Rom 8:9-11
9  You, however, are controlled not by the sinful nature but by the Spirit, if the Spirit of God lives in you. And if anyone does not have the Spirit of Christ, he does not belong to Christ.
10  But if Christ is in you, your body is dead because of sin, yet your spirit is alive because of righteousness.
11  And if the Spirit of him who raised Jesus from the dead is living in you, he who raised Christ from the dead will also give life to your mortal bodies through his Spirit, who lives in you.

1 Cor 3:16
Don't you know that you yourselves are God's temple and that God's Spirit lives in you?

The wording is in both directions, sometimes it sounds like all three are unique, sometimes all are the same thing, or the same as one of the other two.  Sometimes the Holy Spirit sounds like a force, other times like a person.  This is what the Trinity concept is.  God manifests Himself in different ways at different times.  Maybe Christ is as much of God as would fit into human form.

More examples, just like in the Revelation example above, only here both God and Jesus refer to themselves as the same "I AM."

Ex 3:14
14  God said to Moses, "I AM WHO I AM. This is what you are to say to the Israelites: `I AM has sent me to you.'"

John 8:58
58  "I tell you the truth," Jesus answered, "before Abraham was born, I am!"

John 10:27-30
27  My sheep listen to my voice; I know them, and they follow me.
28  I give them eternal life, and they shall never perish; no one can snatch them out of my hand.
29  My Father, who has given them to me, is greater than all; no one can snatch them out of my Father's hand.
30  I and the Father are one."