You seem to forget that Irenaeus was a pupil of Polycarp, who was an auditor of the Apostle John. You live in a society and culture in which something must be written down for it to have credibility. You are seemingly imposing that notion on a time and culture in which many things were passed down orally.
Irenaeus is a very trustworthy source, both in his character and given the fact that he spent considerable time in Rome as an envoy (if I remember correctly). Dionysius, bishop of Corinth (c. 166-174 AD), also writes that both Peter and Paul planted both in Rome and in Corinth, and suffered martyrdom at the same time. And there is no competing traditional account of Peter's life. Peter himself gives evidence of his being in Rome in 1 Peter 5:13. And Papias (bishop of Hieropolis) and Clement of Alexandria both testify that Mark wrote his Gospel at Rome, which Gospel is understood to have been written under the direction and authority of Peter. (See, for example, 1 Pet 5:13) This Clement of Alexandria (c. 150 - 215 AD) also tells us that Peter preached at Rome. Tertullian also refers to "those whom Peter baptized in the Tiber", and tells us that Clement (of Rome) was ordained by Peter at Rome. Remember too that unlike in the case of Paul, there is no evidence of Peter being brought to Rome to stand trial. Nor would Peter, not being a Roman citizen have needed to be brought to Rome to face Nero or to be executed. But there is much evidence that Peter was martyred in Rome. The best explanation of those three facts is that Peter was already in Rome, on his own accord. And there is long-standing and undisputed tradition that Peter went there early, in part to deal with the false teaching of Simon Magus (who apparently had gone to Rome and through his sorcery become so revered that had a statue of himself as a god set up in Rome). St. Cyril (bishop of Jerusalem) testifies to that, and there are other sources for that as well.
-A8
So in other words, Irenaeus has no evidence to include Peter in this statement, but he does have substantial Scriptural and patriarchal evidence to make this statement about Paul.? Right?
Dionysius, bishop of Corinth (c. 166-174 AD), also writes that both Peter and Paul planted both in Rome and in Corinth, and suffered martyrdom at the same time.
Isn't Dionysius untrustworthy as a source? After all he claims that Peter planted the Church in Corinth together with Paul. That's simply not true. Otherwise, Paul would have mentioned him in his letters to Corinth, and Luke in Acts regarding the Corinthians. Isn't it likely that he is confusing "Apollos" in Corinth with Peter.
And there is no competing traditional account of Peter's life. Peter himself gives evidence of his being in Rome in 1 Peter 5:13.
No that's not true either. He is clearly in Babylon there, where Josephus tells us there was a substantial community of Jews [the circumcision to whom Peter dedicated his life to reaching]. If you are going to tell me that "Babylon" was a code word for "Rome", then why didn't Paul use that code word or another one for Rome. What was "Crete" a code word for? Are you going to tell me that Peter at that point in his life was afraid of telling the truth about where he was?
And Papias (bishop of Hieropolis) and Clement of Alexandria both testify that Mark wrote his Gospel at Rome, which Gospel is understood to have been written under the direction and authority of Peter. (See, for example, 1 Pet 5:13) This Clement of Alexandria (c. 150 - 215 AD) also tells us that Peter preached at Rome.
If Clement of Alexandria is correct, Mark might have written his Gospel from Rome, but Peter could have told him to do it years before and/or from some other place. Furthermore, if he is correct that Peter preached in Rome, possibly during his sojourn in Asia Minor, that is a far cry from a 25 year bishopric, that he lived there, that he died there, that he was crucified by Nero.
Tertullian also refers to "those whom Peter baptized in the Tiber", and tells us that Clement (of Rome) was ordained by Peter at Rome.
Well Tertullian here is a problem as well. He was a later writer and where would he get this? By hearsay and rumor again? And here Tertullian is at odds with Irenaeus and Eusebius, who claims that Linus was the first bishop of Rome for 12 years, followed by Anacletus for 12 years, then came Clement as bishop in the 12th year of Emperor Domitian --- 93 AD?????. How was Peter ordaining anybody 25 years after his death? So who's right: Tertullian or Irenaeus or Eusebius? They can't all be right, right? Otherwise why would Peter be ordaining Clement when it should have been Linus that he ordained and 93 AD?????.
Remember too that unlike in the case of Paul, there is no evidence of Peter being brought to Rome to stand trial. Nor would Peter, not being a Roman citizen have needed to be brought to Rome to face Nero or to be executed. But there is much evidence that Peter was martyred in Rome. The best explanation of those three facts is that Peter was already in Rome, on his own accord. And there is long-standing and undisputed tradition that Peter went there early, in part to deal with the false teaching of Simon Magus (who apparently had gone to Rome and through his sorcery become so revered that had a statue of himself as a god set up in Rome). St. Cyril (bishop of Jerusalem) testifies to that, and there are other sources for that as well.
There is only one other source for that and that is Eusebius who was Cyril's [Jerome's] source. So the question is: Where did Eusebius get his information: by oral tradition 300 years from the source? And in that 300 years no one wrote it down?.
You can see the credibility problem that the RCC has here. Those patriarchs are not always trustworthy. That Tradition is often full of holes.
What does the Magisterium really believe regarding Eusebius' report of a 25 year bishopric for Peter in Rome? Does it question it?
How is it possible for Peter to have been in Rome for 25 years and there be no credible written evidence until Eusebius in the 4th century who appears to invent it out of whole cloth, and one filled with holes at that?