Dear Will;
There are really two parts to the issue: (1) Did Peter have Primacy over the other Apostles (Primacy of Peter) ; and (2) Did Christ intend for the office of Peter to be passed on to successive bishops (Succession to the Petrine Office (or the Papacy))
Scriptural support for papal authority and succession.
Part I: Primacy of Peter:
Mt. 16:18 - you are Peter and upon this rock I will build my church
Mt. 16:19 - I will give you the keys of the Kingdom (OT symbol of succession) of the Kingdom; power to bind and loose
Lk 22:32 - Peter’s faith will strengthen his brethren
Jn 21:17 - Jesus gives Peter the flock as chief shepherd
Mk 16:7 - an angel was sent to Peter to announce the resurrection
Lk 24:34 - risen Jesus first appeared to Peter
Acts 1:13-26 - Peter headed the meeting that elected Matthias (also teaches of the succession of the apostolic office)
Acts 2:14 - Peter led the other apostles in preaching on Pentecost
Acts 2:41 - Peter received first converts
Acts 3:6-7 - Peter performed the first miracle after Pentecost
Acts 5:1-11 - Peter inflicted first punishment: Ananias & Saphira
Acts 8:21 - Peter excommunicated the first heretic, Simon Magus
Acts10:44-46 - Peter received revelation to admit Gentiles into the Church
Acts 15:7 - Peter presided over first council in Jerusalem
Acts 15:19 - Peter pronounces first dogmatic decision
Gal. 1:18 - Paul’s visits the chief apostle - Peter - after converting.
Peter’s name always heads the list of apostles: Mt. 10:1-4; Mk 3:16-19; Lk 6:14-16; Acts 1:13.
The Apostles are spoken of as “Peter and his companions”: Lk 9:32; Mk 16:7.
Peter speaks for the apostles: Mt. 18:21; Mk 8:29; Lk 8:45, 12:41 and Jn 6:69.
Peter’s name appears 195 times in the NT, more than all the other Apostles put together.
Part II: Petrine Succession (Christ intended to found a visible church that continued past the death of the Apostles.)
2 Chr 19:11 - high priest over you in everything of the Lord’s
Mal 2:7 - seek instruction from the priest, he is God’s messenger
Eph 2:20 - the Church is built upon the foundation of the apostles and the prophets
Eph 4:11 - God gave some as apostles, others as prophets…
1 Cor. 12:28-29 - God designated in the Church; Apostles, etc.
Acts 1:20 - Decision made to fill apostolic chair left vacant by Judas Iscariot.
Acts 1:25-26 - Matthias elected to fill empty chair.
1 Tim 3:1, 8, 5:17 - qualifications for bishops, priests, deacons in the Church
1 Tim 4:14 - gifts of the priesthood conferred by laying on of hands
1 Tim 5:22 - do not lay hands on anyone too readily
Act 14:23 - presbyters were appointed for every church.
2 Tim 2:2 - what you have heard from me entrust to faithful teachers.
In addition, the early church recognized the authority of Peter and his successors from the early years, as evidenced by the writings of the early church fathers For example, St. Irenaeus in “Against Heresies” stressed that all christians must be united to the Church in Rome in order to maintain the Apostolic Tradition. He presents the teaching as something taken for granted by all orthodox (as opposed to heterodox) christians of his time. (See Jurgens, The Early Church Fathers, #210 and 211.)
In addition to the scriptures, history shows the succession to the See of Rome can be traced all the way back to Peter. The complete list of Popes - 265 in all - can be found in on the internet, in the public library, or in the appendix of a good Catholic study bible.
Hope this helps!
