A beginning of repentance but not fully for he went back to fishing. The cross thing took some time which was part of my point, time.
Are you seriously questioning Peter’s immediate repentance? As soon as that rooster crowed, Peter met the eyes of Jesus and immediately knew what he had done. He didn’t just weep, he bawled like a baby, because he realized that he had denied Jesus, after he insisted that he never would. There was no delay in his repentance. It was instantaneous.
I don’t know if you’re a fisherman or not, but I’ve been fishing since I was big enough to hold onto the branch that my Dad cut from a tree, tied on a piece of throw-line, then added a hook and worm to the end of it. That was well over 50 years ago. Fishing isn’t just something that you do to catch fish, though. Sometimes, I used to go fishing just to think about things that were going on in my life. It’s peaceful and quiet, so it’s a perfect time to pray and contemplate the things that weigh heavy on your heart.
Peter was a fisherman, so his first thought was to do what came natural to him. He did it because it was something familiar to him. It was a kind of coping mechanism to deal with the shock of all that had happened. He needed to think things through, and figure out what he should do next. He certainly didn’t do it because he gave up on Jesus, as you seem to be implying. He didn’t go alone, either. Some of the others went with him, and Jesus knew where to find them. He came to them, while they were there, to comfort them.
Predestination is unconditional election. Infallibility is unconditional for the chair of Peter (on faith and morals).
Catholics don’t believe in predestination. That was a grave error that Calvin taught. Predestination eliminates the possibility of our having free will to choose for ourselves, which is a gift that God gave to the angels and all of mankind. Infallibility has nothing to do with free will. It’s a method used to determine the truth of doctrine, and was given directly to the Church by Jesus. His promise was that the Holy Spirit would lead the Church “into all truth” until Jesus came, again.
Just because the Church teaches the truth, we are still free to choose whether we should accept whatever the Church teaches, or follow our own path. When we choose to follow our own judgement, we run the risk of losing our soul. Some people are willing to take that risk, some are not. Infallibility is a safeguard to let us know what the truth is, so we can make an informed choice. But, in the end, it’s always up to us to choose our own fate. We can always follow our own path if we ‘feel lucky’, but we do it at our own risk. If we’re wrong; we will lose the most precious gift we have.