That's the account of Paul.
1 Corinthians 15: 1-8 (RSV) Now I would remind you, brethren, in what terms I preached to you the gospel, which you received, in which you stand, by which you are saved, if you hold it fast--unless you believed in vain. For I delivered to you as of first importance what I also received, that Christ died for our sins in accordance with the scriptures, that he was buried, that he was raised on the third day in accordance with the scriptures, and that he appeared to Cephas, then to the twelve. Then he appeared to more than five hundred brethren at one time, most of whom are still alive, though some have fallen asleep. Then he appeared to James, then to all the apostles. Last of all, as to one untimely born, he appeared also to me.
You don't have to believe Paul's testimony if you don't want to. He was writing this letter to people who could have checked his story out. You and I can't do that, so we ahve to turn to other tests.
As for the gruesome deaths of the Apostles, those are a part of Church History. They are not well documented but have been handed down for nearly 2000 years.
Shalom.
I see you have a grasp of this argument that nearly equals your grasp of Galileo's political acumen. The proponent I spoke of argued, as I said, that the 500, not the Apostles, were mysteriously put to death by a conspiracy of some sort.
So what about the previous claim that many of them were secretly tracked down and tortured then murdered by some mysterious cabal?
He was writing to the people of Corinth, which was quite some distance from the Holy Land. Hell, people back then believed stories of dog-faced men in Africa and that one-eyed giants inhabited some of the islands of the Mediterranean.
Paul (Saul of Tarsus) never saw Jesus in the flesh or resurrected or in any tangible form. His claim to fame is having a vision of a bright light and hearing a voice just before he went into a 3 day coma while heading to Damascus (Acts 9:3-7).
About AD36, Paul had visited Peter and James in Jerusalem and learned of the legend there. Although he denied it (Gal 1:11-12), his testimony is second hand stories of whatever he heard from them.
While he was a energetic evangelical, Paul is a poor witness for Christ.