Now the apostles and the brothers who were in Judea heard that the Gentiles too had accepted the word of God. So when Peter went up to Jerusalem the circumcised believers confronted him, saying, "You entered the house of uncircumcised people and ate with them." Peter began and explained it to them step by step, saying...
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Here I want to point out not the detail of Peter's explanation, or how baptism of the gentiles came to be. I want to point out to you the language used here, because this is very important. It is describing in unmistakable terms the unique authority given to Peter to not only obey Jesus but to map developments in the faith. This is another piece of the documentation of Peter's papal authority. Peter is fulfilling the pontifical function: "Peter began and explained it to them step by step." Step by step. This is the words of the scripture. Peter is explaining the "new policy" of where he received the authority (his vision from God) and the "step by step" implementation of it, to put in in modern terms.
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18. When they heard this, they stopped objecting and glorified God, saying, "God has then granted life-giving repentance to the Gentiles too."
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They all accepted Peter's unique authority without question. A lot of people read this passage and totally miss the process that is being documented here, focusing only on the literal issue of the preaching to and baptising of the Gentiles. Acts is also recording here the process by which Peter exercised his Christ given authority to interpret what is given to him in vision (personal revelation) and implement it for the faithful, and thus the Church, overall.
How can some Christians not understand the clarity of this? *Argh* Peter is implementing something that "is not in scripture" (it is not something that Jesus said to do before he left them and ascended into heaven). Peter received a vision from God and discerned from it the policy that all faithful will now follow in the Church, which was in this case, the admittance of the Gentiles.