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"Wherefore, adhering faithfully to the tradition received from the beginning of the Christian faith, for the glory of God our Savior, for the exaltation of the Catholic religion, and the salvation of Christian peoples, We teach and define, with the approbation of the Sacred Council, that it is a dogma divinely revealed, that the Roman Pontiff, when he speaks ex cathedra, that is to say, when discharging the functions of Pastor and Doctor of all Christians, by virtue of his supreme Apostolic authority, he defines a doctrine regarding faith and morals to be held by the universal Church, he fully enjoys by the divine assistance promised to him in blessed Peter, the same infallibility which our Divine Redeemer intended His Church should be endowed with for defining doctrine concerning faith or morals; and consequently such definitions of the Roman Pontiff are of themselves irreformable, and not from the consent of the Church.
"If any one presume, which God forbid, to contradict this Our definition, let him be anathema."
The Catechism of Council of Trent, 1962 edition, Baronius Press, Appendix II pp. 542
This unfortunate reality is unavoidable and can be proved using at least three sources:
(a) Paul IV, The Apostolic Constitution: Cum Ex Apostolatus Officio, February 15, 1559:
"6. In addition, [by this Our Constitution, which is to remain valid in perpetuity We enact, determine, decree and define:] that if ever at any time it shall appear that any Bishop, even if he be acting as an Archbishop, Patriarch or Primate; or any Cardinal of the aforesaid Roman Church, or, as has already been mentioned, any legate, or even the Roman Pontiff, prior to his promotion or his elevation as Cardinal or Roman Pontiff, has deviated from the Catholic Faith or fallen into some heresy: the promotion or elevation, even if it shall have been uncontested and by the unanimous assent of all the Cardinals, shall be null, void and worthless. . ."
(b) St. Robert Bellarmine's defense of popes said to have erred in faith.*
*To reference, see: Papal Error? A Defense of Popes said to have Erred in Faith by St. Robert Bellarmine, S.J. Doctor of the Church Translated by Ryan Grant and published by Mediatrix Press
(c) The Impossibility of a Heretical Pope
Argument included as a PDF file here.
(d) The findings of the Cardinal Siri investigation.
Many traditionalist groups who oppose the Second Vatican Council but nonetheless hold that the post-conciliar claimants to the Papacy are nonetheless real Vicars of Christ are simply misrepresenting what the Church has always taught about the Papacy.
One cannot both deny the putatively infallible statements made in the Vatican II documents, solemnly promulgated by Paul VI, and adhere to the Catholic teaching on the infallibility of the Office of Pope.
The inevitable result of both denying Vatican II but holding that the post-conciliar claimants to the Papacy are true popes is a logical inconsistency. The most rational explanation for the crisis that the Church is currently in is that the last six claimants to the Papacy have not been true Vicars of Christ but have in fact, been diabolical usurpers of the Chair of Peter who have desired and seemingly triumphed in leading many souls astray down the road of perdition and sodomy.* One could go further into the devastating effects that the lack of a true Vicar of Christ has had on the world, and one should do research independently to determine the scope of the delusion. However, suffice it to say that there has been nothing but depravity, sin, and hopelessness that has occurred since the overthrow of the rightfully elected pope in 1958. "My people are destroyed for lack of knowledge." (Hosea 4:6)
*The first two of the usurper antipopes, John XXIII and Paul VI, in their surreptitious allowance for birth control, opened the doors for sodomy and every kind of debauchery imaginable under the sun. Artificial contraception opens the doors for overthrowing the primary purpose of the marital act by severing it from the creation of new life...
Pray, fast, and do penance, but most importantly, not give up the hope in the Triumph of the Immaculate Heart of Mary over the whole world. Trust in God to deliver His people from the hands of evildoers, and remember that it is the Church who can act via a willingness to accept a pope should he appear either from heaven, as prophesied by some Catholic mystics including Marie Julie Jehenney, Blessed Anna Maria Taigi, and Elizabeth Canori Mora, or through the actions of men and women in locating a new Vicar to remedy the existing evils. The latter manner of Restoration has been prescribed by great theologians of the Church including St. Robert Bellarmine, and Msgr. Charles Journet, as well as the Dominican historian Father Paul C. Perrotta and has even been cited in the Catholic Encyclopedia of 1908 (see quote below):
"A council in opposition to the pope is not representative of the whole Church, for it neither represents the pope who opposes it, nor the absent bishops, who cannot act beyond the limits of their dioceses except through the pope. A council not only acting independently of the Vicar of Christ, but sitting in judgment over him, is unthinkable in the constitution of the Church; in fact, such assemblies have only taken place in times of great constitutional disturbances, when either there was no pope or the rightful pope was indistinguishable from antipopes. In such abnormal times the safety of the Church becomes the supreme law, and the first duty of the abandoned flock is to find a new shepherd, under whose direction the existing evils may be remedied."
"Councils," The Catholic Encyclopedia, Vol. 4, 1908, p. 426