Posted elsewhere:
There are various arguments used against the Catholic understanding of Matthew 16:18, but all are difficult to argue in favor of.
First of all, Jesus had just gathered all His disciples, and He chose to change Peter's name alone, (Abraham also had his name changed, and he was also called called the "rock" (Isaiah 51:1-2)–one argument used against Peter being the foundation of the Church is how 1 Corinthians 3:11 refers to Jesus as the foundation. This is false precisely because the Apostles are also called twelve foundations (Apocalypse 21:14). What this means is that all authority comes from Christ, the Church itself comes from Christ. Well, Peter's authority comes from Christ. He established these things on Peter, so what's setup on Peter is a foundation on Christ, that does not mean He didn't establish an office which would be the rock on which the Church is built. He gives His keys (Apocalypse 3:7, Apocalypse 1:18) to Peter, as it was prophesied in Isaiah 22:22. Jesus is the Good Shepherd, (John 10:14) but He gives that responsibility to Peter also (John 21-15-17), to say that he then points to Peter and calls him "little rock" (Petros is the masculine version of Petra, the reason Petros is used and not Petra (In Aramaic it was "thou art Kephas and upon this Kephas) is to preserve the play on words, not to indicate that Jesus called him little rock) and point to Himself and says "upon this rock" makes no sense, as right after, He gives Him the keys which He possesses (Apocalypse 3:7, Apocalypse 1:18, Apocalypse 9:1, Apocalypse 20:1, prophesied in Isaiah 22:22), and if you pay attention to the wording, it's indicating an office that is established, not something that passes away with Peter. Understanding what the Keys are and Isaiah 22:22 indicate infallibility, and, again, we literally just have to point to early Popes like Leo the Great to see what Papal infallibility/Supremacy is. And you completely ignore the fact that Luke 22 is about which Apostle is the greatest, and Jesus affirm Peter, and He prays for Him alone. That's not a mistake. There's a strife among the Apostles as to who is the greatest, Jesus responds by saying His Kingdom is not like that of the Gentiles, describing its structure. He says that Satan has desired to sift all the Apostles (plural) but He has prayed for Peter (singular, also the person who has the Keys), alone, that his faith fail not. This is the unfailing faith of the office of Prime Minister of the New Israel. Whatever He binds on Earth, obviously cannot be false in Heaven, therefore, someone in Peter's Chair cannot make a false teaching Ex Cathedra. And do you believe that those who wrote the Bible wrote infallibly? If so, it's not far-fetched to say He can give the Prime Minister infallibility under certain circumstances.