I’m not going to defend Paul’s actions. I have no idea why he did what he did, other than fear, which is the same reason Peter did what he did.
So, when exactly is infallability used? A sermon by the Pope? A Council? A Papal Bull? A book written by the Pope? It seems to me that there is no consistent answer and that sometimes some of these things are considered infallable, while other times they are not considered so. Seems a little too convenient for my taste. Leaves too much wiggle room for there to be any consistency, especially the fact that it takes into mind “intention” of the Pope. The Pope can say anything he wants, meet all the other requirements of being “ex cathedra” and then when he realizes what he said was totally contradictory to what some other Pope or Council said, he just has to say, “Oh, it wasn’t my intent to be infallible.”
Please give examples of where all of this has taken place.
You probably need to do some careful research on infallibility to remove the misunderstandings that you have. I would suggest starting here:
catholic.com/search.asp?query=infallibility
This is taken from one of the articles from that link:
"An infallible pronouncement—whether made by the pope alone or by an ecumenical council—usually is made only when some doctrine has been called into question. Most doctrines have never been doubted by the large majority of Catholics.
Pick up a catechism and look at the great number of doctrines, most of which have never been formally defined. But many points have been defined, and not just by the pope alone. There are, in fact, many major topics on which it would be impossible for a pope to make an infallible definition without duplicating one or more infallible pronouncements from ecumenical councils or the ordinary magisterium (teaching authority) of the Church.
At least the outline, if not the references, of the preceding paragraphs should be familiar to literate Catholics, to whom this subject should appear straightforward. It is a different story with “Bible Christians.” For them papal infallibility often seems a muddle because their idea of what it encompasses is often incorrect.
Some ask how popes can be infallible if some of them lived scandalously. This objection of course, illustrates the common confusion between infallibility and impeccability. There is no guarantee that popes won’t sin or give bad example. (The truly remarkable thing is the great degree of sanctity found in the papacy throughout history; the “bad popes” stand out precisely because they are so rare.)
Other people wonder how infallibility could exist if some popes disagreed with others. This, too, shows an inaccurate understanding of infallibility, which applies only to solemn, official teachings on faith and morals, not to disciplinary decisions or even to unofficial comments on faith and morals. A pope’s private theological opinions are not infallible, only what he solemnly defines is considered to be infallible teaching.
Even Fundamentalists and Evangelicals who do not have these common misunderstandings often think infallibility means that popes are given some special grace that allows them to teach positively whatever truths need to be known, but that is not quite correct, either. Infallibility is not a substitute for theological study on the part of the pope.
What infallibility does do is prevent a pope from solemnly and formally teaching as “truth” something that is, in fact, error. It does not help him know what is true, nor does it “inspire” him to teach what is true. He has to learn the truth the way we all do—through study—though, to be sure, he has certain advantages because of his position."
Those are the general principles. The other articles in that library can help show you what kinds of statements are infallible and which are not. This one may be very helpful to you:
catholic.com/thisrock/2001/0109bt.asp
I hope all of this will be of some help in starting you off in this exploration.