I would check your sources again here, because wherever you got these facts was wildly incorrect.
Irenaeus's list of the bishops of Rome is contained in his Against Heresies III.3. Here's the passage that you are probably referring to:
The blessed apostles, then, having founded and built up the Church, committed into the hands of Linus the office of the episcopate. Of this Linus, Paul makes mention in the Epistles to Timothy. To him succeeded Anacletus; and after him, in the third place from the apostles, Clement was allotted the bishopric...Starts with Linus? Actually it starts with the "blessed Apostles" handing off the office of bishop to Linus. And who were those blessed Apostles? Here's the paragraph right before this one:
Since, however, it would be very tedious, in such a volume as this, to reckon up the successions of all the Churches, we do put to confusion all those who, in whatever manner, whether by an evil self-pleasing, by vainglory, or by blindness and perverse opinion, assemble in unauthorized meetings; [we do this, I say,] by indicating that tradition derived from the apostles, of the very great, the very ancient, and universally known Church founded and organized at Rome by the two most glorious apostles, Peter and Paul; as also [by pointing out] the faith preached to men, which comes down to our time by means of the successions of the bishops. For it is a matter of necessity that every Church should agree with this Church, on account of its preeminent authority -- that is, the faithful everywhere -- inasmuch as the Apostolic Tradition has been preserved continuously by those who are everywhere.Not only does Irenaeus specifically mention Peter and Paul "founded and organized" the Church of Rome, but he says that every Church should agree with the Church there "on account of its preeminent authority". So for a scholar to cite him as evidence against the office of papacy continuing past Peter is, to say the least, seriously odd.
There's much more about Peter in Eusebius than being crucified at the end of his days, particularly in the 2nd and 3rd books.
Eusebius in one part does mention "After the martyrdom of Paul and Peter the first man to be appointed bishop of Rome was Linus." (History of the Church 3:2). He also says, supporting your argument, that "Clement..became the third bishop of Rome". Eusebius therefore seems to be counting from Linus, and not from Peter.
However, Eusebius also calls Linus
"the first after Peter to be appointed Bishop of Rome".And elsewhere:
"Meanwhile At Rome, when Evarestus had completed his eighth year, Alexander took up the Bishopric as fifth successor to Peter and Paul." (4:1)Clearly, whether Eusebius actually called Peter "Bishop of Rome", he envisioned a continuing office of bishop that was established by Peter and continued through Linus, Anacletus and Clement.
Bear in mind that Peter had a far more dignified office than mere bishop--he was an Apostle. We don't call Bush governor of Texas anymore, because he has achieved a higher office--likewise, it may well be that the Fathers did not think of Peter as a bishop, but rather an Apostle. But it is clear, as the above quotes indicate, that the bishopric in Rome (as elsewhere---Antioch, Jerusalem, Alexandria) came directly from his Apostolic office.
How would you address the following, as it pertains to the question of a broken apostolic sucession?
If this is supposed to mean that the Pope is paramount because his authority descends from Peter, there's a little problem. The problem is that the link was broken in the 13th century by Philip Le Bel and Guillaume de Nogart who kidnapped and killed one Pope, assassinated another and then stole the Papacy and moved it, lock stock and Earthly power, to France. Thus began the Avignon captivity. (don't bother to damn the French - they already are)
I believe that there have been truly holy men who have occupied the Papacy since it was returned to Rome. There have also been great sinners. Regardless, I believe the link to Peter has been irrevocably broken.
That was the point that the Cathars had in mind.
The reason I ask is that I think this is quite an important point, and IMO has not been successfully defended against in this thread.
I'm not capable of defending against it because I don't know enough, but from reading your posts, and having confidence in Tantumergo I thought you guys might be able to.