Probably to late for your series but I want to share my thoughts as well....
What the church is:
Composition -
What is it? what are some things that must be in order for a group of individuals to be called a church
Mat 18:20 For where two or three are gathered together in my name, there am I in the midst of them.
There are no other requirements. The church is body of Christ, the members there of. It is in no way, related to any physical structure. An really, I would call the idea of relating it too a building as a lie of the enemy. Because that then locks people into a mentality, that you have to be somewhere in order to meet God.
Actually do consider, how many believe that they walk with God. Simply because they go to a specific location on Sunday or whenever. You do not go to church, you are the church; you are the Temple of the Living God.
What it looks like -
What is its character? What are some of the marks of a church that make it a church?
The Church is a living organism. Always growing, moving, never static.
If you want to see it -
Joh 13:35 By this shall all men know that ye are my disciples, if ye have love one to another.
And being exemplified by the fruit of the Spirit. I will add a note, that this is more then everyone just holding hands and getting along. The meaning here would be at a level to describe the Love and action of God, moving among the members.
Eph 4:15 But speaking the truth in love, may grow up into him in all things, which is the head, even Christ:
Eph 4:16 From whom the whole body fitly joined together and
compacted by that which every joint supplieth, according to the effectual working in the measure of every part, maketh increase of the body unto the edifying of itself in love.
Everyone has a part, there is no sitting in a chair/pew while watching someone else mediate between us and God. Also, this is NOT referring to some volunteer activity like mowing the lawn, doing projection or child care. Any true function, would have a place in a meeting of only 2 people. Weather it's on a beach, at house, or in a prison cell.
How does it function? -
Eph 4:11 And he gave some, apostles; and some, prophets; and some, evangelists; and some, pastors and teachers;
Eph 4:12 For the perfecting of the saints, for the work of the ministry, for the edifying of the body of Christ:
Eph 4:13
Till we all come in the unity of the faith, and of the knowledge of the Son of God, unto a perfect man, unto the measure of the stature of the fulness of Christ:
This may sound crazy....... Jesus is the head of the church/body, NOT the pulpit or ministry. Verse 13 says they are here
"Until we all come in". Their function is to help the body function (v12), and then join the body in that. In other words, their role is to work themselves out of job.
Purpose -
Why does it exist, what is its purpose, how does it fulfill this purpose in the local body?
The simplest answer, is to edify the rest of the body and reflect/effect Him to the world.
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Jesus told Peter that on this rock I will build my church. I will really have to study that more. I wish I knew more of the life of Peter outside of Acts.
Mat 16:16 And Simon Peter answered and said, Thou art the Christ, the Son of the living God.
Mat 16:17 And Jesus answered and said unto him, Blessed art thou, Simon Barjona:
for flesh and blood hath not revealed it unto thee, but my Father which is in heaven.
Mat 16:18 And I say also unto thee, That thou art Peter, and upon this rock I will build my church; and the gates of hell shall not prevail against it.
Jesus was not referring to Peter, when he said this. HE was referring to the act of hearing God (v17); receiving revelation from the Father. It is upon that rock (the ability to hear God's Truth), that the church is setup. For no man can come unto the Father, unless the Spirit calls Him.
Joh 6:65 And he said, Therefore said I unto you, that no man can come unto me, except it were given unto him of my Father.
Salvation can not be had by mental accent (James 2:19, 1Cor 2:11). It requires hearing and properly responding to God's revelation. There is no other way to be a part of the body/church.
And it is upon this rock (the ability to hear God's Truth), that the gates of hell can not prevail.
Plus I want to give others a chance to chime in before I hijack your thread
You forgot to hijack it!