The Primacy of St. Peter

 Writing on this topic, Fr. John Laux writes in "Catholic Apologetics Book IV":

The protest contention [that the powers bestowed on St. Peter by Christ in Matthew 16 were personal and not permanent] is manifestly wrong; for it is clear from the words with which Our Lord conferred the Primacy on Peter that it was not a personal privilege, such as the power of working miracles or the gift of tongues, but a permanent institution, necessary for the very existence of the Church. 

The Church which Christ founded is to endure till the end of time, and Peter, according to the promise of Christ, is to be the rock foundation of that Church, giving it unity and strength, and securing it against the gates of hell. But the foundation must last as long as the ediface lasts which is built upon it. Hence the power and authority that made Peter the Rock of the Church must remain intact for all time; his Primacy must be perpetuated in the only possible way, that is, by transmission through a continuous line of successors.

Fr. Laux continues:

The Bishops of Rome have always claimed to be the successors of St. Peter, the heirs of his Primacy. No other See in Christendom ever made such a claim. St. Peter ruled the Church of Antioch before he went to Rome, yet Antioch never disputed the right of Rome to the unique dignity of Apostolic Primacy. Ephesus and Smyrna and the other Churches of Asia Minor over which St. John, the Beloved Disciple, presided long after Peter's death never contested the claim of Rome.

This is further shown through a study of history. St. Clement of Rome, the third Successor of St. Peter, governed the Church and exercised the Primacy of the Papal Office for the first recorded time in 96 AD when he dispatched a powerful letter to the Corinthians. This fact of his intervention and jurisdiction is remarkable since the Apostle St. John was himself still alive and presiding over the Church in Ephesus. 

Jesus summoned his Twelve disciples and gave them authority over unclean spirits to drive them out and to cure every disease and every illness. The names of the Twelve Apostles are these: first, Simon called Peter, and his brother Andrew; James, the son of Zebedee, and his brother John; Philip and Bartholomew, Thomas and Matthew the tax collector; James, the son of Alphaeus, and Thaddeus; Simon the Cananean, and Judas Iscariot who betrayed Jesus. Jesus sent out these Twelve after instructing them thus,“Do not go into pagan territory or enter a Samaritan town. Go rather to the lost sheep of the house of Israel. As you go, make this proclamation: ‘The Kingdom of heaven is at hand.’”   Matthew 10:1-7

We notice St. Peter is always listed first in every list of the apostles in the Gospels - Judas Iscariot is always listed last. This is one point showing the Primacy of Peter - he is higher than all of the other disciples. Peter's name is mentioned 155 times. All of the other eleven disciples combined are listed only 130 times.

Furthermore, we see in John 21:15-19, the discourse between Peter and Jesus. In this discourse, Jesus says, "Simon, son of John, do you love me more than these [the other disciples]?" Peter says "yes". In that conversation, Peter professes his faith in Christ three times to make up for the three times he denied our Lord during His Passion. Unlike Judas, who lost his soul because he refused to ask God's mercy for his actions, Peter is forgiven because he seeks forgiveness.

In the Acts of the Apostles, Peter is the spokesmen and always speaks for the group. People would lay in the streets just wanting Peter's shadow to fall on them (Acts 5:15). We see that St. Peter was given a higher authority than the other disciples. St. Peter was the first pope - the one that Christ founded His Church upon (Matthew 16:18-20).