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You cannot disprove apostolic succession.

Rather, what you cannot prove is apostolic succession, that the original apostles were succeeded by popes and prelates distinctively titled "hiereus”="priests" who were to engage in a uniquely sacerdotal acts as their primary ordained function, offering flesh and blood as a expiatory sacrifice and dispensing it to the people in order to obtain spiritual and eternal life.

The Scriptural facts are that there are no apostolic successors named (like for James: Acts 12:1,2) after Judas, who was to chosen maintain the original 12 (Rv. 21:14) - thus only one was chosen. Nor was any elected by voting, versus casting lots (no politics). (Acts 1:15ff)

And nowhere does the Holy Spirit ever refer to NT pastors (presbuteros=senior/elder or episkopos= superintendent/overseer, these being one office: Titus 1:5-7) as “hiereus” or “archiereus” (priest or high priest).

What occurred is that "presbuteros" in Greek (presbyter in Latin) was translated into English as "preost," and then "priest," but which also became the word used for "hierus" ("sacerdos" in Latin), losing the distinction the Holy Spirit made by never distinctively giving NT presbuteros the distinctive title hiereus.

More.

And instead of dispensing bread as part of their ordained function, which NT pastors are never described as doing in the life of the church, and instead the primary work of NT pastors is that of prayer and preaching. (Act 6:3,4) "Preach the word; be instant in season, out of season; reprove, rebuke, exhort with all longsuffering and doctrine." (2 Timothy 4:2)

And which is what is said to "nourish" the souls of believers, and believing it is how the lost obtain life in themselves. (1 Timothy 4:6; Psalms 19:7;Acts 15:7-9)

And turning to history as the church became progressively deformed in becoming like the empire in which it was found, w have the findings of such RCs as,

Klaus Schatz [Jesuit Father theologian, professor of church history at the St. George’s Philosophical and Theological School in Frankfurt] in his work, “Papal Primacy ,” pp. 1-4, finds:

“New Testament scholars agree..., The further question whether there was any notion of an enduring office beyond Peter’s lifetime, if posed in purely historical terms, should probably be answered in the negative.

That is, if we ask whether the historical Jesus, in commissioning Peter, expected him to have successors, or whether the authority of the Gospel of Matthew, writing after Peter’s death, was aware that Peter and his commission survived in the leaders of the Roman community who succeeded him, the answer in both cases is probably 'no.”

If we ask in addition whether the primitive church was aware, after Peter’s death, that his authority had passed to the next bishop of Rome, or in other words that the head of the community at Rome was now the successor of Peter, the Church’s rock and hence the subject of the promise in Matthew 16:18-19, the question, put in those terms, must certainly be given a negative answer...” (page 1-2)

If one had asked a Christian in the year 100, 200, or even 300 whether the bishop of Rome was the head of all Christians, or whether there was a supreme bishop over all the other bishops and having the last word in questions affecting the whole Church, he or she would certainly have said no." (page 3, top)

“We probably cannot say for certain that there was a bishop of Rome [in 95 AD]. It is likely that the Roman church was governed by a group of presbyters from whom there very quickly emerged a presider or ‘first among equals’ whose name was remembered and who was subsequently described as ‘bishop’ after the mid-second century.” (Schatz 4).

Schatz additionally states, Cyprian regarded every bishop as the successor of Peter, holder of the keys to the kingdom of heaven and possessor of the power to bind and loose. For him, Peter embodied the original unity of the Church and the episcopal office, but in principle these were also present in every bishop. For Cyprian, responsibility for the whole Church and the solidarity of all bishops could also, if necessary, be turned against Rome." — Papal Primacy [Collegeville, Minnesota: The Liturgical Press, 1996], p. 20)

• Roman Catholic scholar William La Due (taught canon law at St. Francis Seminary and the Catholic University of America) on Cyprian:

....those who see in The Unity of the Catholic Church, in the light of his entire episcopal life, an articulation of the Roman primacy - as we have come to know it, or even as it has evolved especially from the latter fourth century on - are reading a meaning into Cyprian which is not there." (The Chair of Saint Peter: A History of the Papacy [Maryknoll, New York: Orbis Books, 1999], p. 39

• Catholic theologian and a Jesuit priest Francis Sullivan, in his work From Apostles to Bishops (New York: The Newman Press), examines possible mentions of “succession” from the first three centuries, and concludes from that study that,

“the episcopate [development of bishops] is a the fruit of a post New Testament development,” and cannot concur with those (interacting with Jones) who see little reason to doubt the notion that there was a single bishop in Rome through the middle of the second century:

Hence I stand with the majority of scholars who agree that one does not find evidence in the New Testament to support the theory that the apostles or their coworkers left [just] one person as “bishop” in charge of each local church... - — Francis Sullivan, in his work From Apostles to Bishops , pp. 221,2 22,224

Roman Catholic [if liberal and critical] Garry Wills, Professor of History Emeritus, Northwestern U., author of “Why i am a Catholic,” states,

"The idea that Peter was given some special power that could be handed on to a successor runs into the problem that he had no successor. The idea that there is an "apostolic succession" to Peter's fictional episcopacy did not arise for several centuries, at which time Peter and others were retrospectively called bishops of Rome, to create an imagined succession. Even so, there has not been an unbroken chain of popes. Two and three claimants existed at times, and when there were three of them each excommunicating the other two, they all had to be dethroned and the Council of Carthage started the whole thing over again in 1417." — WHAT JESUS MEANT, p. 81

• American Roman Catholic priest and Biblical scholar Raymond Brown (twice appointed to Pontifical Biblical Commission), finds,

“The claims of various sees to descend from particular members of the Twelve are highly dubious. It is interesting that the most serious of these is the claim of the bishops of Rome to descend from Peter, the one member of the Twelve who was almost a missionary apostle in the Pauline sense – a confirmation of our contention that whatever succession there was from apostleship to episcopate, it was primarily in reference to the Puauline tyupe of apostleship, not that of the Twelve.” (“Priest and Bishop, Biblical Reflections,” Nihil Obstat, Imprimatur, 1970, pg 72.)

Eamon Duffy (Pontifical Historical Commission, Professor of the History of Christianity at the University of Cambridge, and former President of Magdalene College) states: “Self-consciously, the popes began to model their actions and their style as Christian leaders on the procedures of the Roman state”. — Eamon Duffy notes (“Saints and Sinners”, ©2001 edition)

• Eastern Orthodox scholarship also adds voice to this,

It is illuminating to understand that even some very illustrious Roman Catholic theologians today recognize that the Papacy as it now exists is of late origin. W. DeVries admits, “...throughout the first ten centuries Rome never claimed to have been granted its preferred position of jurisdiction as an explicit privilege” (Orthodoxy, Roman Catholicism and Anglicanism by Methodios Fouyas, p. 70). Avery Dulles considers the development of the Papacy to be an historical accident. “The strong centralization in modern Catholicism is due to historical accident. It has been shaped in part by the homogeneous culture of medieval Europe and by the dominance of Rome, with its rich heritage of classical culture and legal organization” (Models of the Church by Avery Dulles, p. 200) http://www.orthodoxresearchinstitute.org/articles/ecumenical/maxwell_peter.htm

Then you have the manner of preservation of faith and "unbroken succession that preceded the Reformation:

Referring to the schism of the 14th and 15th centuries, Cardinal Ratzinger observed,

"For nearly half a century, the Church was split into two or three obediences that excommunicated one another, so that every Catholic lived under excommunication by one pope or another, and, in the last analysis, no one could say with certainty which of the contenders had right on his side. The Church no longer offered certainty of salvation; she had become questionable in her whole objective form--the true Church, the true pledge of salvation, had to be sought outside the institution.

"It is against this background of a profoundly shaken ecclesial consciousness that we are to understand that Luther, in the conflict between his search for salvation and the tradition of the Church, ultimately came to experience the Church, not as the guarantor, but as the adversary of salvation. (Joseph Cardinal Ratzinger, head of the Sacred Congregation of the Doctrine of the Faith for the Church of Rome, “Principles of Catholic Theology,” trans. by Sister Mary Frances McCarthy, S.N.D. (San Francisco: Ignatius, 1989) p.196). http://www.whitehorseinn.org/blog/2012/06/13/whos-in-charge-here-the-illusions-of-church-infallibility/)