For no one can lay any foundation other than what is being laid, which is Jesus Christ.1 Corinthians 3:11

The idea of the papacy is not found anywhere in Scripture. Catholics do try very, very hard, however, to make it seem as though it is.

Peter's Confession

13 When Jesus came to the area of Caesarea Philippi, he asked his disciples, "Who do people say that the Son of Man is?" 14 They answered, "Some say John the Baptist, others Elijah, and others Jeremiah or one of the prophets." 15 He said to them, "But who do you say that I am?" 16 Simon Peter answered, "You are the Christ, the Son of the living God." 17 And Jesus answered him, "You are blessed, Simon son of Jonah, because flesh and blood did not reveal this to you, but my Father in heaven! 18 And I tell you that you are Peter, and on this rock I will build my church, and the gates of Hades will not overpower it. 19 I will give you the keys of the kingdom of heaven. Whatever you bind on earth will have been bound in heaven, and whatever you release on earth will have been released in heaven." 20 Then he instructed his disciples not to tell anyone that he was the Christ.Matthew 16:13-20

Catholics claim that the "rock" in verse 18 is Peter, and that Christ built His church upon Peter, the first Pope. Aside from the general argument of the rest of Scripture, there are all sorts of reasons why Catholics are badly, badly mistaken in their interpretation.

Refuting the Catholic Heresy

The Greek

The Greek word for "Peter" is petros, which is masculine in gender, and refers to a small stone. The root of petros is another Greek word, petra, which is feminine and refers to a huge rock (such as would be used in the foundation of a building). Here is verse 18 with the Greek words inserted:

And I tell you that you are Petros, and on this petra I will build my church, and the gates of Hades will not overpower it.Matthew 16:18

The two words have different meanings, and in context, they refer to two different things. So, if Christ wasn't referring to Peter as the rock upon which the Church would be built, who or what was He referring to?

I'm no Greek scholar, but I'm told that the Greek does not allow for Peter to be the subject of Christ's reference. The actual subject, as indicated by the language itself, it Peter's statement back in verse 16: "You are the Christ, the Son of the living God." Specifically, Christ is referring to Himself as the rock upon which He would build the Church. In fact, this is what Scripture teaches more explicitly elsewhere.

Tradition

The Roman Catholic church claims to be the true church instituted by Christ Himself, firmly founded in tradition. However, when Calixtus I first tried to use the above passage to support the formation of the papal office, the original Greek was used to shut him down. Notice two things here: It 1) shows that the papacy can not be traced all the way back to Peter, as we can even see someone trying to start the papacy, and 2) the early church (tradition) rejected the modern Catholic interpretation of Peter's confession. Obviously the Catholic church only adheres to tradition when it's convenient.

Peter refers to Christ as the rock

4 So as you come to him, a living stone rejected by men but chosen and priceless in God's sight, 5 you yourselves, as living stones, are built up as a spiritual house to be a holy priesthood and to offer spiritual sacrifices that are acceptable to God through Jesus Christ. 6 For it says in scripture, "Look, I lay in Zion a stone, a chosen and priceless cornerstone, and whoever believes in him will never be put to shame." 7 So you who believe see his value, but for those who do not believe, the stone that the builders rejected has become the cornerstone, 8 and a stumbling-stone and a rock to trip over. They stumble because they disobey the word, as they were destined to do.1 Peter 2:4-8

Christ is the rock

For no one can lay any foundation other than what is being laid, which is Jesus Christ.1 Corinthians 3:11

and all drank the same spiritual drink. For they were all drinking from the spiritual rock that followed them, and the rock was Christ.1 Corinthians 10:4

Peter did not have special authority

Peter did not exercise authority over the Church as the Pope claims authority to do. Instead, Peter was obedient to the Church:

Now when the apostles in Jerusalem heard that Samaria had accepted the word of God, they sent Peter and John to them.Acts 8:14

Notice that Peter did not send anyone anywhere, but rather he himself was sent by the other apostles.

1 So as your fellow elder and a witness of Christ's sufferings and as one who shares in the glory that will be revealed, I urge the elders among you: 2 Give a shepherd's care to God's flock among you, exercising oversight not merely as a duty but willingly under God's direction, not for shameful profit but eagerly. 3 And do not lord it over those entrusted to you, but be examples to the flock.1 Peter 5:1-3

In the above verse, Peter refers to himself as merely "your fellow elder," not as having any special position or authority in the Church. Notice also that Peter urges the elders not to lord their authority over others. Historically, the apostate Catholic church and its heretical leader have not heeded Peter's advice.

Paul corrected Peter

Paul would not knowingly speak against someone in authority:

3 Then Paul said to him, "God is going to strike you, you whitewashed wall! Do you sit there judging me according to the law, and in violation of the law you order me to be struck?" 4 Those standing near him said, "Do you dare insult God's high priest?" 5 Paul replied, "I did not realize, brothers, that he was the high priest, for it is written, 'You must not speak evil about a ruler of your people.'"Acts 23:3-5

Certainly, if Peter held an even higher position (pope, head of the church), Paul would not have criticized him. But Paul did criticize Peter:

11 But when Cephas came to Antioch, I opposed him to his face, because he had clearly done wrong. 12 Until certain people came from James, he had been eating with the Gentiles. But when they arrived, he stopped doing this and separated himself because he was afraid of those who were pro-circumcision. 13 And the rest of the Jews also joined with him in this hypocrisy, so that even Barnabas was led astray with them by their hypocrisy. 14 But when I saw that they were not behaving consistently with the truth of the gospel, I said to Cephas in front of them all, "If you, although you are a Jew, live like a Gentile and not like a Jew, how can you try to force the Gentiles to live like Jews?"Galatians 2:11-14

Peter probably never went to Rome

Catholics claim that Peter was the Bishop of Rome. However, there is no historical evidence for this. In Romans 16, Paul lists more than 25 people to be greeted and thanked, yet Peter is not listed anywhere. Wouldn't Paul note the Peter if he really was Bishop of Rome?

Peter was not treated as Pope

Paul says in Galatians 1:12 that he received the Gospel directly from Christ, yet it is nowhere said that Peter did. Why, if Peter held such an office as the papacy, did he not have at least the same honor given Paul?

In Matthew 20:20-28, the subject of special position comes up. Christ rebukes the apostles when there is tension over this issue, saying, "You know that the rulers of the Gentiles lord it over them, and those in high positions use their authority over them. It must not be this way among you! Instead whoever wants to be great among you must be your servant, and whoever wants to be first among you must be your slave." Notice that Christ did not give Peter (the supposed foundation of the church) any special recognition. He also said that the first among them should be their servant, not their ruler.