The Apostle Paul

The Apostle Paul, who was used of the Lord to give instructions to the Church, did not once refer to a successor to himself or the other apostles as such. When he was nearing the end of his course, he sent from Miletus to Ephesus to have the elders come to meet him. There he gave them a farewell address and if there could be any place where Paul would have mentioned a successor it should have been there. He said: "And now, behold, I know that ye all, among whom I have gone preaching the kingdom of God, shall see my face no more.... Take heed therefore unto yourselves, and to all the flock, over the which the Holy Ghost hath made you overseers, to feed the church of God.... For I know this, that after my departing shall grievous wolves enter in among you, not sparing the flock. Also of your own selves shall men arise, speaking perverse things, to draw away disciples after them." Acts 20:25-30.
Paul as an apostle had appointed elders to look after the flock; they were not apostles, and in view of his departure, he told them what would come in after he left the scene. Some of them would speak perverse things to draw disciples after them. Perhaps these pretended to be his successors, for the Lord later addresses the Church at Ephesus commending them for trying some who said they were apostles, but were not (Rev. 2:2).