1st Argument by Jimmy Akin:
According to many current sedevacantists, Pius XII was the last valid pope. He died in 1958, which was 53 years ago.
Here is where the argument gets interesting: In order to be pope, under current canon law, one must be elected by the college of cardinals. In order to be a member of the college of cardinals, one must be appointed by the pope. In order for the pope to appoint you, he must be alive.
If the last valid pope died in 1958, that would seem to mean that no cardinals have been validly appointed since then. How many cardinals are alive today who were appointed before 1958?
The longest-serving cardinal at present is Eugenio Sales, who wasn’t appointed until 1969. If his elevation to the cardinalate was invalid, and so were all subsequent elevations due to a lack of valid popes, then it would appear that the college of cardinals now has no members. With no valid members, it would seem impossible for there to be another validly elected pope.
That would be odd.
It would certainly seem to be contrary to the will of Christ who, in the words of Vatican I, willed “that blessed Peter should have perpetual successors in the primacy over the whole Church.” If Christ really wills that there be an ongoing series of successors then one would think he would keep the Church from getting into a position where it is impossible to elect any more successors.
So do we have a good argument here, from Vatican I, after all? An argument that deals a death-blow to a major current form of sedevacantism?
2nd Argument (Quoted by Nishant):
Cardinal Billot says: "God may allow that a vacancy of the Apostolic See last for a while. He may also permit that some doubt be risen about the legitimacy of such or such election. However, God will never allow the whole Church to recognize as Pontiff someone who is not really and lawfully. Thus, as long as a pope is accepted by the Church, and united with her like the head is united to the body, one can no longer raise any doubt about a possible defective election."
3rd Argument of Pope Benedict XIV (Quoted by Nishant):
it suffices Us to be able to state that a commemoration of the supreme pontiff and prayers offered for him during the sacrifice of the Mass is considered, and really is, an affirmative indication which recognizes him as the head of the Church, the vicar of Christ, and the successor of blessed Peter, and is the profession of a mind and will which firmly espouses Catholic unity. This was rightly noticed by Christianus Lupus in his work on the Councils: "This commemoration is the chief and most glorious form of communion" (tome 4, p. 422, Brussels edition).
This view is not merely approved by the authority of Ivo of Flaviniaca who writes: "Whosoever does not pronounce the name of the Apostolic one in the canon for whatever reason should realize that he is separated from the communion of the whole world" (Chronicle, p. 228); or by the authority of the famous Alcuin: "It is generally agreed that those who do not for any reason recall the memory of the Apostolic pontiff in the course of the sacred mysteries according to custom are, as the blessed Pelagius teaches, separated from the communion of the entire world" (de Divinis Officiis, bk. 1, chap.
1) The first two arguments establish the impossibility of sedevacantism.
2) The 3rd argument establishes the schismatic nature of the sedevacantist position.
3) All three in combination establish that all the various sedevacantist theories are total hallucinations.