It starts with a look at the importance of dealing with this issue:
"If anyone, therefore, shall say that the Blessed Peter the Apostle was not appointed the Prince of the Apostles and the visible head of the whole Church militant, or that the same directly and immediately received from the same our Lord Jesus Christ a primacy of honor only, and not of true and proper jurisdiction; let him be anathema." - First Vatican CouncilThe Catholic Church upped the ante. This cannot be a minor issue of disagreement.
This is followed by good analysis of the Matthew 16:13-20 passage.
I particularly like the reference to Luke 22:24 ("A dispute also arose among them, as to which of them was to be regarded as the greatest.")
"Instead of acknowledging Peter's privileged position, they are debating who, in fact, is the greatest among them. If Jesus had indeed elevated Peter, it seems that the other apostle's missed the point, Peter didn't invoke it, and Christ didn’t reiterate it--even though this was surely a teaching moment."And, of course, 1 Peter 5:1-2.
The next section is excellent analysis from Paul's writings.
A nice point on the belief that Peter was Bishop of Rome from an early date:
"if he were, then he abandoned Paul in prison, because Paul wrote that at his first trial everyone abandoned him."The final section (and the clincher for me), is an analysis of the first council (in Jerusalem).
"It was not Peter who lead the Jerusalem council, it was James, as is seen by the imperative mode of his speech. The letter that is written does not carry the imprimatur of Peter, but of all the apostles."