The church says that after Peter was Linus (67-76), then Anacletus (76-88) then Clement (88-97), Evaristus (97-105), Alexander I (105-115), Sixtus I (115-125) and so on down to modern times. They base their infallibility on this unbroken line from Peter.
These claims cannot, however, be proven either by scripture or by history. First, scripture never says Peter was ever the bishop of Rome or that he ever even served there and the absence of any mention of Peter in Paul’s letter to the Romans is quite telling in this case. Especially if the Roman myth is correct and Peter has already been there some 20 years by this time not to even mention him. Or that Peter is not mentioned as one who visited Paul while imprisoned there as well. It simply did not happen. Nor is there any evidence Rome was the churches governmental seat any time before the fourth century. And since Peter was by decree of the Spirit of God the apostle to the Jews he would have most certainly left during the command for all Jews to leave Rome issued in AD 50 by Claudius. Finally, we have the scriptural evidence that for the first 15 chapters of Acts, Peters ministry was centered in the Middle East, Jerusalem, Judea, Samaria, Galilee and Antioch, and not in Rome as the myth seems to teach.