Roman Catholicism claims the papacy as its foundation. According to the Catholic Church, the doctrine of the papacy was understood and universally accepted as early as the time of Peter:
"At open variance with this clear doctrine of Holy Scripture as it has been ever understood by the Catholic Church are the perverse opinions of those who, while they distort the form of government established by Christ the Lord in his Church, deny that Peter in his single person, preferably to all the other Apostles, whether taken separately or together, was endowed by Christ with a true and proper primacy of jurisdiction; or of those who assert that the same primacy was not bestowed immediately and directly upon blessed Peter himself, but upon the Church, and through the Church on Peter as her minister....For none can doubt, and it is known to all ages, that the holy and blessed Peter, the Prince and Chief of the Apostles, the pillar of the faith and foundation of the Catholic Church, received the keys of the kingdom from our Lord Jesus Christ, the Saviour and Redeemer of mankind, and lives presides and judges, to this day and always, in his successors the Bishops of the Holy See of Rome" (First Vatican Council, session 4, chapters 1-2)
Different Catholics interpret these claims of the First Vatican Council in different ways. Some Catholics will argue that the concept of the papacy that was understood and accepted in the earliest generations involved universal jurisdiction, so that the differences between how modern Catholics and the most ancient Catholics viewed Peter and the bishops of Rome would be minor. Other Catholics claim, instead, that the earliest Christians wouldn't have associated a concept like universal jurisdiction with Peter and the earliest Roman bishops, and they maintain that the modern view of the papacy developed more gradually. Some Catholics even go as far as to claim that there's no need to show that a concept like universal jurisdiction was intended by Jesus and the apostles. They may argue for the papacy on the basis of philosophical speculation or personal preference, or they may claim that no argument is needed for the doctrine.
Catholics who take that last sort of approach are abandoning the battlefield without admitting defeat. Any belief could be maintained on such a basis. If we're going to accept the papacy just because it seems to produce more denominational unity than other systems of church government, because our parents were Catholic, or for some other such inconclusive reason, then we have no publicly verifiable case to make for the doctrine. My intention in these posts is to address some of the popular arguments of those who attempt to make a more objective case for the papacy.
Those who argue that a seed form of the papacy existed early on, one that wasn't initially associated with universal jurisdiction, would need to demonstrate that such a seed form of the doctrine did exist. And they would need to demonstrate that the concept of universal jurisdiction would eventually develop from that seed. It wouldn't be enough to show that the development of universal jurisdiction is possible. We don't believe that something is true just because it's possible. If we're supposed to accept a papacy with universal jurisdiction on some other basis, such as the alleged authority of the Catholic hierarchy that teaches the concept, then an objective case will have to be made for the supposed authority of that hierarchy.
If there had been a papacy in the first century that was recognized as a distinct office, we would expect it to be mentioned in much the same way that offices such as bishop and deacon are mentioned. We wouldn't expect Roman Catholics to have to go to passages like Matthew 16 and John 21 to find alleged references to a papacy if such an office of universal jurisdiction existed and was recognized during the New Testament era. Instead, we would expect explicit and frequent references to the office, such as in the pastoral epistles and other passages on church government.
That's what we see with the offices of bishop and deacon. Not only are the offices mentioned (Acts 20:17, Philippians 1:1), but we also see repeated references to their appointment (Acts 14:23, Ephesians 4:11, Titus 1:5), their qualifications (1 Timothy 3:1-13, Titus 1:5-9), their discipline (1 Timothy 5:19-20), their responsibilities (Ephesians 4:12-13, Titus 1:10-11, James 5:14, 1 Peter 5:1-3), their reward (1 Timothy 5:17-18, 1 Peter 5:4), their rank (1 Corinthians 12:28), the submission due them (1 Timothy 2:11-12), etc. If there was an office that was to have jurisdictional primacy and infallibility throughout church history, an office that could be called the foundation of the church, wouldn't we expect it to be mentioned explicitly and often? But it isn't mentioned at all, even when the early sources are discussing Peter or the Roman church. In the New Testament, which covers about the first 60 years of church history (the prophecies in Revelation and elsewhere cover much more), there isn't a single Roman bishop mentioned or named, nor are there any admonitions to submit to the papacy or any references to appointing Popes, determining whether he's exercising his infallibility, appealing to him to settle disputes, etc. When speaking about the post-apostolic future, the apostles are concerned with bishops and teachers in general (Acts 20:28-31, 2 Timothy 2:2) and submission to scripture (2 Timothy 3:15-17, 2 Peter 3:1-2, Revelation 22:18-19), but don't say a word about any papacy.
Craig Keener, citing Jaroslav Pelikan, comments that "most scholars, both Roman Catholic and Protestant, concur that Peter died in Rome but doubt that Mt 16:18 intended the authority later claimed by the papacy (Pelikan 1980: 60)" (A Commentary on the Gospel of Matthew [Grand Rapids, Michigan: William B. Eerdmans Publishing Company, 1999], n. 74 on p. 425). The Roman Catholic scholar Klaus Schatz comments:
"There appears at the present time to be increasing consensus among Catholic and non-Catholic exegetes regarding the Petrine office in the New Testament….The further question whether there was any notion of an enduring office beyond Peter’s lifetime, if posed in purely historical terms, should probably be answered in the negative. That is, if we ask whether the historical Jesus, in commissioning Peter, expected him to have successors, or whether the author of the Gospel of Matthew, writing after Peter’s death, was aware that Peter and his commission survived in the leaders of the Roman community who succeeded him, the answer in both cases is probably 'no.'…If we ask in addition whether the primitive Church was aware, after Peter’s death, that his authority had passed to the next bishop of Rome, or in other words that the head of the community at Rome was now the successor of Peter, the Church’s rock and hence the subject of the promise in Matthew 16:18-19, the question, put in those terms, must certainly be given a negative answer." (Papal Primacy [Collegeville, Minnesota: The Liturgical Press, 1996], pp. 1-2)
What's said of Peter in Matthew 16 and John 21 is said of other people in other passages. Other people are rocks upon whom the church is built (Ephesians 2:20), other people have the keys of the kingdom that let them bind and loose and open and shut (Matthew 18:18, 23:13), and other people are shepherds of the church (Acts 20:28, 1 Peter 5:2). Just as Peter is given a second name, so are other people (Mark 3:17). Peter is called "Peter" prior to the events of Matthew 16 (John 1:42), and we can't know whether he was given the name as a result of Matthew 16 or, instead, Jesus' choice of imagery in Matthew 16 was shaped by a name Peter was already given for another reason.
Peter is singled out in Matthew 16 and John 21, but his being singled out doesn't suggest jurisdictional primacy. We could speculate that Peter is singled out in these passages because he's supposed to fulfill the roles in these passages in a greater way than other people, but such a speculation can't be proven. Other people are singled out in other passages, but we don't conclude that those people were Popes. Even if Peter was singled out because he was to fulfill these roles (rock and shepherd) in a greater way than anybody else, he wouldn't need to be a Pope in order to fulfill these roles in a greater way than other people. And he wouldn't need to have successors in that role.
So, if Peter isn't singled out in Matthew 16 and John 21 because he was being made a Pope, then why was he singled out?
In Matthew 16, he's probably singled out because he singles himself out. He's the one who answered Jesus' question. Similarly, John and James are singled out in Mark 10:35-40 because they were the ones who initiated the discussion with Jesus, not because they were being given some sort of primacy.
In John 21, Peter probably is singled out because he was the one in need of restoration. Peter was the one who denied Jesus three times and thus needed to reaffirm his love for Jesus three times. Since the other apostles didn't deny Jesus as Peter did, it would make no sense for Jesus to approach them the way He approached Peter. Similarly, Jesus treats Thomas (John 20:26-29), John (John 21:20-23), and Paul (Acts 9:1-15) differently than He treats the other apostles. But nobody would assume that Thomas, John, or Paul therefore has jurisdictional primacy or that such a primacy was passed on to a succession of bishops.
Catholics sometimes argue for a papacy by interpreting Matthew 16 in light of Isaiah 22:20-22. But whatever relevance Isaiah 22 would have to Matthew 16, it would have relevance for Matthew 23, Luke 11, and other passages that use such imagery as well. And any Catholic appeal to Isaiah 22 would have to be a partial appeal, not a complete parallel, since a complete parallel wouldn't favor the claims of Roman Catholicism. God is the one who gives the key in Isaiah 22, so an exact parallel would put Jesus in the place of God, not in the place of the king. So, if Jesus is God and Peter is the prime minister, then who is the king? Some church official with more authority than Peter? What about Isaiah 22:25? Should we assume that Popes can "break off and fall", and that the keys of Matthew 16 can eventually pass to God Himself (Revelation 3:7) rather than to a human successor? If Catholics only want to make a general appeal to Isaiah 22, without making an exact parallel, then how can they claim that papal authority is implied by the parallel? Why can't the Isaiah 22 background convey a general theme of authority without that authority being of a papal nature?
Paul refers to "apostles" (plural) as the highest rank in the church (1 Corinthians 12:28, Ephesians 2:20), and he names Peter second among three reputed pillars of the church (Galatians 2:9). The most natural reading of the Biblical evidence is to see Peter as a highly reputed pillar of the church who had equal rank, equal jurisdiction, with the other apostles. He could be said to have had some types of primacy in some contexts, and the same could be said of other apostles and early church leaders, but there's no reason to think that papal authority was one of those types of primacy or that such authority was passed on exclusively to a succession of Roman bishops.
There is no papacy in the New Testament. It's not there explicitly or implicitly. This "clear doctrine of Holy Scripture" that the First Vatican Council refers to isn't even Biblical, much less clearly Biblical. Roman Catholics assume that a papacy is implied in some New Testament passages, but that assumption can't be proven and is unlikely.