The following excerpt from Wayne Grudem's Systematic Theology explains the qualifications for being an apostle.
a. Qualifications of an Apostle: The two qualifications for being an apostle were (1) having seen
Jesus after his resurrection with one’s own eyes (thus, being an “eyewitness of the
resurrection”), and (2) having been specifically commissioned by Christ as his apostle.4
The fact that an apostle had to have seen the risen Lord with his own eyes is indicated by Acts
1:22, where Peter said that person to replace Judas “must become with us a witness to his
resurrection.” Moreover, it was “to the apostles whom he had chosen” that “he presented himself
alive after his passion by many proofs, appearing to them during forty days” (Acts 1:2–3; cf.
4:33).
Paul makes much of the fact that he did meet this qualification even though it was in an unusual
way (Christ appeared to him in a vision on the road to Damascus and appointed him as an
apostle: Acts 9:5–6; 26:15–18). When he is defending his apostleship he says, “Am I not an
apostle? Have I not seen Jesus our Lord?” (1 Cor. 9:1). And when recounting the people to
whom Christ appeared after his resurrection, Paul says, “Then he appeared to James then to all
the apostles. Last of all, as to one untimely born, he appeared also to me. For I am the least of
the apostles, unfit to be called an apostle” (1 Cor. 15:7–9).
These verses combine to indicate that unless someone had seen Jesus after the resurrection
with his own eyes, he could not be an apostle.
The second qualification, specific appointment by Christ as an apostle, is also evident from
several verses. First, though the term apostle is not common in the gospels, the twelve disciples
are called “apostles” specifically in a context where Jesus is commissioning them, “sending them
out” to preach in his name: And he called to him his twelve disciples and gave them authority
over unclean spirits, to cast them out, and to heal every disease and every infirmity. The names
of the twelve apostles are these....These twelve Jesus sent out charging them, “...preach as you
go, saying, “The kingdom of heaven is at hand.”’ (Matt. 10:1–7)
Similarly, Jesus commissions his apostles in a special sense to be his “witnesses...to the end of
the earth” (Acts 1:8). And in choosing another apostle to replace Judas, the eleven apostles did
not take the responsibility on themselves, but prayed and asked the ascended Christ to make
the appointment:
“Lord, who knows the hearts of all men, show which one of these two you have chosen to
take the place in this ministry and apostleship from which Judas turned aside....” And they
cast lots for them, and the lot fell on Matthias; and he was enrolled with the eleven
apostles. (Acts 1:24–26)
Paul himself insists that Christ personally appointed him as an apostle. He tells how, on the
Damascus Road, Jesus told him that he was appointing him as an apostle to the Gentiles: “I
have appeared to you for this purpose, to appoint you to serve and to bear witness...delivering
you from the people and from the Gentiles—to whom I send you” (Acts 26:16–17). He later
affirms that he was specifically appointed by Christ as an apostle (see Rom. 1:1; Gal. 1:1; 1 Tim.
1:12; 2:7; 2 Tim. 1:11). b. Who Were Apostles? The initial group of apostles numbered
twelve—the eleven original disciples who remained after Judas died, plus Matthias, who
replaced Judas: “And they cast lots for them, and the lot fell on Matthias; and he was enrolled
with the eleven apostles” (Acts 1:26). So important was this original group of twelve apostles, the
“charter members” of the office of apostle, that we read that their names are inscribed on the
foundations of the heavenly city, the New Jerusalem: “And the wall of the city had twelve
foundations, and on them the twelve names of the twelve apostles of the Lamb” (Rev. 21:14).
We might at first think that such a group could never be expanded, that no one could be added
to it. But then Paul clearly claims that he, also, is an apostle. And Acts 14:14 calls both Barnabas
and Paul apostles: “when the apostles Barnabas and Paul heard of it....” So with Paul and
Barnabas there are fourteen “apostles of Jesus Christ.”
Then James the brother of Jesus (who was not one of the twelve original disciples) seems to be
called an apostle in Galatians 1:19: Paul tells how, when he went to Jerusalem, “I saw none of
the other apostles except James the Lord’s brother.”6 Then in Galatians 2:9 James is classified
with Peter and John as “pillars” of the Jerusalem church. And in Acts 15:13–21, James, along
with Peter, exercises a significant leadership function in the Jerusalem Council, a function which
would be appropriate to the office of apostle. Furthermore, when Paul is listing the resurrection
appearances of Jesus he once again readily classifies James with the apostles: Then he
appeared to James then to all the apostles. Last of all, as to one untimely born, he appeared
also to me. For I am the least of the apostles, unfit to be called an apostle, because I persecuted
the church of God. (1 Cor. 15:7–9)
Finally, the fact that James could write the New Testament epistle which bears his name would
also be entirely consistent with his having the authority which belonged to the office of apostle,
the authority to write words which were the words of God. All these considerations combine to
indicate that James the Lord’s brother was also commissioned by Christ as an apostle. That
would bring the number to fifteen “apostles of Jesus Christ” (the twelve plus Paul, Barnabas, and
James).
Were there more than these fifteen? There may possibly have been a few more, though we
know little if anything about them, and it is not certain that there were any more. Others, of
course, had seen Jesus after his resurrection (“Then he appeared to more than five hundred
brethren at one time,” 1 Cor. 15:6). From this large group it is possible that Christ appointed
some others as apostles—but it is also very possible that he did not. The evidence is not
sufficient to decide the issue.