In answering this question, I have been helped by four articles written by Pastor John MacArthur, where he described the requirements for an apostle: chosen by God, appointed by Jesus, witness of the risen Christ, and unique ministry duties. He demonstrates that these requirements proves that there cannot be apostles today. These articles are a good example of systematic theology done well, where the combined weight of the testimony of many passages of Scripture proves a point that no single passage establishes on its own.
I’m going to summarize each argument with screenshots of appropriate passages, with a brief discussion under each.
First, an apostle must have been chosen by God. Paul called himself “an apostle of Christ Jesus by the will of God” (2 Cor 1:1; see also, among others, Eph 1:1; 2 Tim 1:1). God had chosen Paul, according to Paul’s testimony, “before I was born” (Gal 1:15). A claim to be an apostle is an enormous one, claiming as it does extrabiblical insight into the purposes and will of God.
Secondly, an apostle must have been appointed by Jesus. This is also a huge claim. The original twelve apostles could substantiate this with their personal testimony of the incarnate Christ choosing them (Luke 6:13). Paul, even though he was “untimely born” (1 Cor 15:8), was also chosen personally by Jesus (Acts 9:15; Gal 1:15-16), and learned the gospel from his own lips (Gal 1:12).
Thirdly, an apostle must have been a witness of the risen Christ. This requirement is seen in the deliberation of the eleven remaining disciples, after Judas’s suicide, as they endeavored to see whom God had chosen to replace him: they said that such a one “have accompanied us during all the time that the Lord Jesus went in and out among us, beginning from the baptism of John until the day when he was taken up from us” (Acts 1:21-22). And even though Paul was not among that number, he claimed that “last of all," Jesus had “appeared also to me” (1 Cor 15:8; cf. Acts 9:1-8).
And finally, an apostle had unique ministry duties. These began during Christ’s ministry, as Mark 3:14-15 testifies. The apostles were to be “with” Jesus, and “to preach and have authority to cast out demons” (see also Matt 28:19-20; Rom 1:4-5). Jesus also gave them the ability to heal diseases (Luke 9:1-20).
Can anyone today claim that God has chosen him, that Jesus has personally appointed him to be an apostle, and that he has seen the risen and glorified Jesus with his own eyes? Anyone can make such claims, of course, but that is a far cry from verifying them. Such claims would have to be substantiated by miracles, exorcisms, and healings. And where are these things found? I answer: nowhere. The original apostles actually did heal diseases (see Acts 3:1-10, 12-16 for one example), unlike the fraudulent claims of modern-day hucksters like Benny Hinn and Kenneth Copeland.
There are other supporting arguments for the cessation of the office of apostle—ones which are not sufficient to make the case on their own but which add weight when considered along with the requirements of an apostle. For instance, there are only instructions in the NT for choosing elders and deacons. We are given no guidance on selecting apostles (or prophets or evangelists, for that matter). Then there is the fact that, over and over again, the apostles are called “the twelve.” They needed a replacement after Judas’s suicide to make the number twelve again (Acts 1:24-26), and John, in his vision, saw the New Jerusalem, which not only had twelve gates with the numbers of the twelve tribes of Israel, but also “twelve foundations, and in them the names of the twelve apostles of the Lamb” (Rev 21:14).
The testimony of the totality of the New Testament, including the four requirements for apostleship, is that the gift of apostle has ceased.
I look forward to discussion and further digging into the Scriptures on this topic!