OK, picking up where I left off:
Paul's third proof of his apostleship is his whereabouts since his conversion.
I did not immediately consult with anyone; nor did I go up to Jerusalem to those who were apostles before me, but I went away into Arabia, and returned again to Damascus. (Galatians 1:16-17): Upon his conversion, Paul's first thought was not to rush off to Jerusalem and meet with the Twelve. He didn't collude with them, or with anyone else. Instead, the first place he went was Arabia. This is not the Arabian penninsula as we would understand it today; rather, he meant the kingdom of Nabatea, which corresponds roughly with present-day Jordan and the Sinai Penninsula. Its capital was Petra, that fantastic ancient city carved into the cliff faces.
Why was Paul in Arabia? He doesn't say. Some people suggest that he was evangelizing the Gentiles that lived there. Others suggest that he spent time alone reflecting upon his conversion. Still others say he was learning his theology through a direct revelation of Christ himself. What is important, however, is not what Paul was doing, but where he was. If he was in Arabia, he was not consulting with the other apostles in Jerusalem.
and returned again to Damascus (17): Luke writes of Paul's time in Damascus in Acts 9: 19-25. However, Luke doesn't talk about Paul's trip to Arabia, so it gives the impression that Paul was in Damascus for a few days before being compelled to escape. Maybe this was Paul's first stay in Damascus; maybe it was his whole time there (interrupted by a trip to Arabia). There's no frame of reference to say for sure. However, Paul and Luke are writing about Paul's conversion and early ministry for two different purposes. Luke is writing about Paul as a new believer, and the reactions of local Christians and Jews to his newfound faith. Paul, on the other hand, is saying that he was in Damascus early on, and not consulting with the other apostles in Jerusalem.
Then after three years I went up to Jerusalem to visit Cephas and remained with him fifteen days. But I saw none of the other apostles except James the Lord's brother (18-19): Now, three years after his conversion at least, Paul goes to Jerusalem. This is probably the trip to Jerusalem that Luke describes in Acts 9:26-30.
Again, it appears difficult to harmonize: Paul says that he met only with Peter and James, while Luke says he tried to join the other Christians, but they were afraid of him until Barnabas took him to the apostles and vouch for him. However, if we were to give both authors the benefit of the doubt, the two accounts can reasonably be harmonized. Perhaps the "apostles" Luke has in mind are Peter and James specifically.
In fact, the real controversy in verse 19 has to do with whether Paul is naming James amongst the apostles. The Greek here is ambiguous, and could mean, to paraphrase, "I didn't see any of the other apostles except James" (implying he was an apostle), or "I didn't see any of the other apostles, but I did see James" (implying he wasn't); or even some sort of middle ground, such as, "I didn't see any of the other apostles, unless you count James, and I saw him." My belief is that Paul is naming James amongst the apostles. He would not be the only person outside of the Twelve or Paul named an apostle (Barnabas is also so named, in Acts 14:14). Also, the general details of his life are similar to Paul's: he was initially an unbeliever, who saw the risen Jesus (1 Corinthians 15:7), and then rose to leadership within the church.
In any case, contrast the length of time Paul spends in Jerusalem (15 days) with the time he spends in Damascus and Arabia (more than 3 years). He came to Jerusalem to get acquainted with Peter. The Judaizers who attempted to discredit Paul did so in part by claiming we was at best a secondary apostle who got his theology and commission from the Twelve. Although Paul and Peter almost certainly shared their individual experiences with the risen Christ during their visit, the two weeks Paul spent in Jerusalem is hardly enough time to establish him as a follower of Peter.
(In what I am writing to you, before God, I do not lie!) (20): This is serious business, if Paul is literally swearing an oath that he is telling the truth. Swearing falsely in God's name was a violation of the third commandment.