What is the evidence upon which they base this claim?
Historically, there are only "traditions" that are poorly substantiated. There are no Bible verses that ever support directly any such possibility.
There is only one verse that has been (mis)interpreted to say that Peter was writing from Rome.
This is 1 Pet. 5:12. Peter's letter is to those who were "the pilgrims of the Dispersion in Pontus, Galatia, Cappodicia, Asia and Bythynia," (1 Pet. 1:1). He closes by saying (5:12) "She who is in Babylon, elect together with you, greets you, and so does Mark my son."
Roman Catholics contend that "Babylon" is a code name that Peter used for Rome, as did John when he wrote Revelation 14:8, 16:19 etc. But is this a correct interpretation, and did Peter ever go to Rome?
The most succinct critique of this Roman Catholic proposition I have read is by Criswell in his commentary on this verse in his footnote to The Believer's Bible (NIV). I give it in full.
1 Peter 5:13
Peter is probably alluding to the Babylon on the Euphrates. a part of that Eastern world where he lived and did his work, rather than Rome (with Babylon being utilized as a cryptic word). Evidence for this position includes the following:
(1) There is no evidence that Rome was ever called Babylon until after the writing of the Book of Revelation in A.D. 90-96, many years after Peter's death. [Note 2]
(2) Peter's method and manner of writing are not apocalyptic. On the contrary, Peter is a man plain of speech, almost blunt, who would not interject such a mystical allusion into his personal explanations and final salutation.
(3) Babylon is no more cryptic than Pontus, Asia, or the other places mentioned when Peter says the elect in Babylon send greetings to the Jews of the Dispersion in Pontus, Galatia, Cappadocia, Asia, and Bithynia.
(4) Babylon, no longer a great world capital in the time of Peter, was still inhabited by a colony of people, mostly Jews, many of whom Peter befriended and won to Christ.
(5) A study of the chronology of Peter's travels argues for "Babylon" to be the Babylon on the Euphrates. Such a study reveals these significant points:
(a) In A.D. 40, three years after Paul's conversion and subsequent travels into Arabia, Peter was still in Jerusalem. Around that time, he [Paul] made his missionary journey through the western part of Judea to Lydda, Joppa, Caesarea, and back to Jerusalem (Acts 9-1 1 ).
(b) Imprisoned under Herod Agrippa I, he was miraculously delivered by the angel of the Lord (Acts 12). Peter was probably still in the vicinity of Palestine when Herod Agrippa I died (Acts 12:17, 20-23). The date, according to Josephus, was the fourth year of the reign of Claudius, c. A.D. 45. In A.D. 54, soon after Paul visited Peter again in Jerusalem (Gal: 2), Peter returned the visit by going to Antioch where Paul was working and where the famous interview between the two occurred (Gal. 2:11-14).
(c) From A.D. 54 to c. A.D. 60, Peter apparently made an extensive missionary journey (or journeys) throughout the Roman provinces of the East, taking his wife with him (1: Cor. 9:5). During their travels in Pontus, Galatia, Cappadocia, Asia and Bithynia, Peter and his wife remained in the Orient, never entering Rome. One can verify this by the last chapter of the epistle to the church at Rome, written c. A.D. 60, in which Paul salutes 27 persons, never mentioning Peter. It would seem that Paul did not send him greetings simply because Peter neither was there nor ever had been. Those who hold that Peter governed a church at Rome must face the fact of Paul's omission of Peter's name: Had Peter been in Rome, the omission would have been a gross insult. Furthermore, it had been agreed at the Jerusalem Conference that Peter should go to the Jews and Paul to the Gentiles. The church at Rome was Gentile (Rom. 1:13) [MB - but seemingly there were Jews there - Rom. 3:1-8 - although Paul MAY have referred to Jews in these verses so that he could show that ALL men were guilty], and Paul was eager to go where no other apostle had been (Rom: 15:20; 2 Cor. 10:15, 16). Since he wrote his Roman epistle to the people at Rome, Paul's desire to witness to that city would be inexplicable had Peter been there at the time, or had he ever spent a number of years there. Neither while Paul was under Roman imprisonment from about A.D. 60 to 63, when he wrote four letters - Ephesians, Philippians, Colossians, and Philemon - nor shortly before his death, when he wrote his final letter to young Timothy, did he mention Peter. In his letters he mentioned many fellow Christians who were in Rome, but he stated clearly in 2 Tim. 4:11 that only Luke was with him.
NOTE 1 - "ON THIS ROCK" AND "THE KEYS OF THE KINGDOM"
These two verses are also examined by Criswell as follows;
16:18 Matthew provides a play on the Greek words in this crucial quotation from Jesus: "You are petros [a small throwable stone], and on this petra [the feminine form of petros, which indicates an immovable ledge of rock] I will build My church." Historic interpretations include the following:
(1) Peter is the rock upon which the church fs to be constructed, since Jesus was probably speaking Aramaic, in which language He word simply would have been Cephas in both instances.
(2) Peter is addressed but Jesus indicates, possibly by means of gesture, that the Christ is the petra upon which the church will be built (cf. 1 Cor. 3:11).
(3) Peter is addressed and told that his confession (v. 16) is the petra upon which Jesus will build His church. Either of the latter two positions is plausible; both of these take into account the word changes written by Matthew. A promise of perpetuity is given even in the face of the devastating ravages of death, whose gates seemingly enclose all things. "Hades" cannot destroy or "hold down" the church.
16:19 Though v. 18 is difficult, v. 19 has been even more fiercely debated. However, nothing in either verse suggests the possibility that Peter was given authority to forgive men of their sins. The words "bind" and "loose" are rabbinic terms meaning "to forbid" and "to permit." "Keys" were the symbol of knowledge or the fruit of the scribal or teaching office (cf. Luke 11;52 in which "key" refers to knowledge). Peter and those with him are given the "keys of the kingdom," i.e., the gospel of Christ. The use of those keys will build the church. Peter did precisely this at Pentecost (Acts 2:14), at Samaria (Acts 8:14), and for Cornelius the Gentile (Acts 10). The expressions "will be bound in heaven" and "will be loosed in heaven" are examples in Greek of the periphrastic future perfect passive construction and should, therefore, be translated "will have been bound already" and "will have been loosed already in heaven. In other words, Peter's pronouncement of "binding" or "loosing" is dependent upon what heaven has already willed, rather than earth's giving direction to heaven.
NOTE 2 - THE LATEST DATING OF THE NT
(1) I do not accept that ANY part of the New Testament was written after AD 70 when Jerusalem fell to the Romans. Such a cataclysmic event would surely have been recorded by any one of the writers. Revelation is the last book written and is very symbolic but does it not in any credible sense refer to the destruction of Jerusalem. The destruction it describes is that of "Babylon" which is symbolic of Rome.
(2) In Revelation 13:18, John gives the gematrial number for Nero Caesar which is 666. However, the Greek words - Nero Kesar - gives only 616. It is the Hebrew word - Neron Kesar - that gives 666 exactly, and would have been understood by his Jewish readers. It would appear that John is referring to Nero specifically, and if this is so, then he must have written this between AD 65 when he started the persecutions and AD 68 when Nero died. Revelation was therefore written before the Fall of Jerusalem in AD 70. (See "True Science Agrees with the Bible" p.433.)
(3) John Robinson wrote a book in which he contends that the failure of the NT writers to even hint at the destruction of Jerusalem is good evidence that they were all writing before that great event.
NOTE 3 - THE PRIORITY OF THE FOUNDING OF THE BRITISH CHURCH
There are documents declaring that priority must be given to British church leaders as they were the first nation to officially adopt the Christian faith. This took place as early as 58 AD by Bran, the father of Caractacus, the feared British warrior, who was held as hostage in Rome. (See "True Science Agrees with the Bible" p.140, 142 - Section; British Christianity.)
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