The Roman Catholic church claims special authority for itself on account of the alleged fact that the disciple Peter was the first Bishop of Rome and that Christ had given Peter special powers that Peter passed on to later bishops of Rome. The Bishop of Rome is these days referred to as the Pope, which simply means "father". So there are 3 claims there in need of validation.
1). There is no mention in the NT that Peter was ever in Rome. It was Paul who went to Rome according to the NT. But could Peter have followed on later? If so, such an important event would surely have been noted somewhere at the time in the 1st century. The Catholic Encyclopedia can rustle up just 3 alleged 1st century references:
"Earlier still is Clement of Rome writing to the Corinthians, probably in 96, certainly before the end of the first century. He cites Peter's and Paul's martyrdom as an example of the sad fruits of fanaticism and envy. They have suffered "amongst us" he says, and Weizsaecker rightly sees here another proof for our thesis.
The Gospel of St. John, written about the same time as the letter Clement to the Corinthians, also contains a clear allusion to the martyrdom by crucifixion of St. Peter, without, however, locating it (John 21:18, 19).
The very oldest evidence comes from St. Peter himself, if he be the author of the First Epistle of Peter, of if not, from a writer nearly of his own time: "The Church that is in Babylon saluteth you, and so doth my son Mark" (1 Peter 5:13). That Babylon stands for Rome, as usual amongst pious Jews, and not for the real Babylon, then without Christians, is admitted by common consent (cf. F.J.A. Hort, "Judaistic Christianity", London, 1895, 155).
It should be obvious that these are all weak reeds to lean upon.
What did Clement mean by "amongst". That it meant "in Rome" is just one interpretation. Since Clement was bishop of Rome, however, it may be this selfsame sly allusion that gave rise to the later belief that Peter reached Rome. As Bishop of Rome, Clement would have an obvious interest in fostering such a myth.
I pass over the second "reference" in polite silence.
The third reference asserts that there were no Christians in Babylon at the time. But there certainly were Jews and the famous Babylonian Talmud eventually emerged as the product of their deliberations. So it is entirely plausible that Peter did go there in an attempt to make converts and had some success. So this passage too is no proof of anything.
I would have entertained the idea that "Babylon" was symbolic if the reference had come from a sometimes gnostic writer like St. John but Peter writes a perfectly straightforward book of instructions. I think we must take him at his word. He went to Babylon, not Rome.
2). Special powers conferred? The basis for this claim is the passage in Matthew 16:18. "And Jesus answering said to him: Blessed art thou, Simon Bar-Jona: because flesh and blood hath not revealed it to thee, but my Father who is in heaven. And I say to thee: That thou art Peter; and upon this rock I will build my church, and the gates of hell shall not prevail against it. And I will give to thee the keys of the kingdom of heaven. And whatsoever thou shalt bind upon earth, it shall be bound also in heaven: and whatsoever thou shalt loose on earth, it shall be loosed also in heaven."
Transliterating the relevant Greek of the original we get: "ou ei petros kai epi tautee tee petra oikodomeeso mou teen ekkleesian". That shows that Christ was using two different words for Peter and the rock upon which he was to build his "church'. He was making a distinction, not an equation. I go into more detail about the Greek passage here
An issue seldom addressed, however, is that Christ spoke Aramaic, not Greek. So what we read in Matthew is itself a translation. So what was Christ most likely to have been saying in Aramaic?
Alfred Persson has done the most extensive exploration of the Aramaic background to the text but he really rambles on so I will try to summarize: He points out that "petros" is the Aramaic word for "firstborn" but that it was also known at the time (educated Israelites at the time spoke Greek, as indeed did educated Romans) that the same word in Greek meant "rock". So Jesus was using that known double meaning to make a point vivid.
What point? What was the rock upon which he would build his group of followers? That is no mystery at all. There are numerous references in the NT which equate Jesus's TEACHINGS with a rock -- e.g. Matthew 7:24; 1 Corinthians 10:4. So Jesus expected his teachings to form the foundation of a new group. He was certainly right about that! To encourage his followers, Jesus then goes on to say that the wisdom he imparts is very special indeed. It will give his followers entry into the kingdom of heaven. So the new group will be a privileged one indeed. Orthodox teachings among the Israelites at the time foresaw a resurrection to life on earth, not a transformation into spirit beings.
But what about: "And whatsoever thou shalt bind upon earth, it shall be bound also in heaven: and whatsoever thou shalt loose on earth, it shall be loosed also in heaven."? Which comes first, the chicken or the egg? Catholics claim that the passage gives Christians on earth the power to control events in Heaven. But that is surely absurd. Christ was surely saying that his teachings are an accurate guide to what has already been bound or loosed in Heaven.
3). But say we ignore all of the above and concede that Peter was given some special power. Where is there any statement or evidence in Christ's words that this power could be passed on? There is none. So all three of the Roman claims are mere assertions with no obvious truth value.