John Reumann best states the problem

The Farfa Project
“The Petrine ministry has been at the center of the modern ecumenical discussion, for it concerns the very question of the unity of the church. The International Bridgettine Centre in Farfa Sabina has seriously undertaken a study of the theological, historical, and dogmatic issues that underlie the issues of Christian unity dealing with the role of unity as exercised by the Bishop of Rome.

“This current work represents only part of the work that the Centre has been engaged in during these past ten years. The texts published in this volume give a ‘snapshot’ of the situation when they were presented during the series of colloquia, study groups, and consultations that have taken place. Therefore there has been no attempt to update the bibliographical information. It is important to note that the work is still ‘a work-in-progress.’”.

This is from the Preface of How Can the Petrine Ministry Be a Service to the Unity of the Universal Church? James F. Puglisi, Editor, Grand Rapids, MI and Cambridge, U.K.: William B. Eerdmans Publishing Company, ©2010, pgs. 49-78.

It is a work described as “twenty-one forward-looking essays on the papal office by an assortment of theologians, canonists, ecumenists, ecclesiologists, sociologists, and Scripture experts from diverse backgrounds, including Catholic, Orthodox, Anglican, Lutheran, Methodist, and Reformed [review here]. They examine the conditions under which the papacy might one day be re-received by Christian church bodies worldwide—not as an autocratic monarchy, but, rather, as the unifying agency for a diverse yet cohesive universal church” (from the Back Cover).

John Reumann, the Ministerium of Pennsylvania Professor Emeritus of New Testament and Greek at The Lutheran Theological Seminary at Philadelphia (LTSP), defines the problem as well as any that I’ve seen:

Biblical and patristic studies make clear that historically a gap occurs at the point where it has been claimed “the apostles were careful to appoint successors in” what is called “this hierarchically constituted society,” specifically “those who were made bishops by the apostles . . .,” an episcopate with an “unbroken succession going back to the beginning.” [64] For that, evidence is lacking, quite apart from the problem that the monepiscopacy replaced presbyterial governance in Rome only in the mid-or late second century.[65] It has been noted above how recent treatments conclude that in the New Testament no successor for Peter is indicated.

They not only “suggest” or “posit that” “no successor for Peter is indicated”, but rather, they “conclude” this. The footnotes clarify:

64. Lumen gentium 20 (Flannery trans., Vatican Council II [Collegeville, MN: Liturgical Press, 1975], pp. 371-372; Abbot trans., Documents of Vatican II [New York: Guild Press, America Press, Association Press], pp. 39-40, “the episcopate in a sequence running back to the beginning”). Cited are Iren. Adv. Haer. 3,3,1 = PG 7:848; Tertullian, Praescr. Haer. 32 = PL 2:52f., and Ignatius of Antioch passim.

65. Gnilka 2002, p. 225. Ignatius had no “succession; bishop and presbyter correspond to Christ and apostles, not successors to the apostles (p. 223); the “succession lists” in Rome were of presbyters and bishops (pp. 242-50). (Referring to Gnilka, Joachim. Petrus und Rom. Das Petrusbild in den ersten zwei Jahrhunderten. Freiburg/Basel/Wien: Herder, 2002).

In short, there is a basic conflict between the actual history of the period, and what is claimed in official dogmatic statements of Vatican II. (Much less the discrepancies with Vatican I). This is a major difficulty and embarrassment for the “infallible Magisterium”. I am grateful that it is being discussed publicly, by major theologians.

This work is described as “a rare glimpse into a high-level discussion that should be appreciated by anyone interested in the future of the ‘one, holy, catholic, and apostolic church.’” Reuman and the other writers (and I’ll try to provide selections in the coming days) are taking as their starting point the disjunction between the historical evidence and the doctrinal statements. It is unfortunate, however, that these individuals are writing in order to find some kind of theological justification for a “petrine ministry.”

Carl Trueman was partially correct. Not only do we need a “thoughtful confessional book” on Roman Catholicism; but we need to have evangelical and confessionally-minded Christians involved in these types of discussions at the highest levels. Those who are prone to remembering that the papacy has very recently been called “antichrist” and who are open to suggesting, not that the papacy be “reformed,” but that it be consigned to the ash heap of history.