Let's Look at the Facts
First and foremost, there is tremendous confusion concerning the early "lists" of the bishops of Rome, and for good reason. Different sources give different renderings. Why? As simple as it may sound, the reason is easily discovered: no one really cared for the first century of the history of the church at Rome. All the lists come from at the earliest many decades later, and show a concern that did not arise until the Church as a whole began struggling with heresy and began formulating concepts of authority to use against heretics. But in those first decades, even into the middle of the second century, no one was particularly concerned about who the bishop of Rome was. Why? Because no one had the concepts that Rome now presents as "ancient." No one thought the bishop of any one church was above any other, or that the bishop of Rome was somehow invested with any particular authority.
No Monarchical Episcopate
What's more, there is a fatal historical fact that is overlooked consistently by Roman Catholic apologists. Joseph F. Kelly in his The Concise Dictionary of Early Christianity (The Liturgical Press, 1992), p. 2, notes,
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