Just what is an "apostle" and what qualifies one as such? Good questions. If you consider the reasons Paul might be one, you'd think having writings included as the bulk of New Testament is good enough. "Sure, he may not be part of the twelve, but that doesn't mean he isn't an apostle; neither does it mean he wasn't used by God!" Some might concede as much, but not many.
The word "apostle" and "apostleship" are derived from Greek words. We're going to define these words and consider their occurrences. We're going to use Vine's Complete Expository Dictionary of Old and New Testament Words.
apostolos is, "lit, 'one sent forth (apo, 'from,' stello, 'to send')." What's interesting about the term is that it's actually applied to Jesus, which makes sense because he was sent from the Father. Jesus was given authority from the Father to preach his message.
Next, we would say we see the term specifically used for the 12 that Jesus called and named as such. Here, again, we see a direct giving of authority. First, the Father to Jesus; second, Jesus to the 12. We saw in a previous chapter ("The Twelve"). what exact requirements were necessary for selecting a replacement for Judas. Vine's is truthful in this point.
The twelve disciples chosen by the Lord for special training were so called, Luke 6:13; 9:10. Paul, though he had not 'companied with' the Twelve 'all the time' of His earthly ministry, and hence was not eligible for a place among them, according to Peter's description of the necessary qualifications, Acts 1:22. Paul was commissioned directly by the Lord Himself, after His ascension, to carry the gospel to the Gentiles.
Vine's admits that Paul wasn't a companion with Jesus or the twelve, siding the Peter's conclusion. Now we're tasked for the rest of our journey with determining whether or not Paul was actually "commissioned directly. by the Lord Himself, after His ascension, to carry the gospel to the Gentiles." This task will require we suspend our pre-conceived notions and traditions. Only an unbiased review will yield an honest conclusion. That's another study in a different chapter. For now, let's see what else Vine's says.
The word has also a wider reference. In Acts 14:4, 14, it is used of Barnabas as well as of Paul; in Rom. 16:7 of Andronicus and Junias. In 2 Cor. 8:23 (RV, margin) two unnamed brethren are called "apostles of the churches"; in Phil. 2:25 (RV, margin) Epaphroditus is referred to as "your apostle." It is used in 1 Thess. 2:6 of Paul, Silas, Timothy, to define their relation to Christ.
Vines also notes that apostole is, "a sending, a mission," and "signifies an apostleship, Acts 1:25; Rom 1:5; 1 Cor. 9:2; Gal. 2:8." I must now concede and admit that, in general, if we survey the entire New Testament writings, we see the word indeed has a "wider reference." However, we must pause and consider a few things. Here are some logical considerations given all we know thus far:
- Who says the term has a "wider reference"? Peter narrowly defined an apostle as one who companied with Jesus and the 12 from John's baptism until the ascension and was a direct eye-witness to the bodily resurrection. It would seem reasonable the term is narrowly defined by the short chain of succession we see concluding with the selection of Matthias: First, the Father to Jesus; Second, Jesus to the 12; Third, the Father selects Matthias.
- The "wider reference" we see used of the term is of Paul and his associates in his own epistles, with exception of Luke's account. Perhaps Paul named himself an apostle and selected his own apostles. This would certainly "widen" the reference beyond what Peter and the one who taught him (Jesus) had in mind.
- We must therefore seriously consider whether or not Paul had actually met Jesus Christ on that road to Damascus, and in light of doing this we must consider whether or not that truly qualifies him as a bearer of the title "apostle" given the limited application given by Peter.
So while I may concede for a time the term seems to have "wider reference," we're investigating whether or not it's safe to widen the definition merely because our suspect Paul does.
An error in this conversation is assuming Peter simply made up the qualifications he gave the others in Acts chapter 1. However, I would remind the reader that Peter also said the office at that time had to be filled in accordance with prophetic Scripture. We simply can't assume Peter was being a tyrant and making things up, nor can we assume they were wrong in selecting Matthias. If we read the Psalms Peter refers to, it becomes apparent Peter was correct in having a limited idea of the term.
 John 17:3; Hebrews 3:1
 Matthew 10:2-4, Mark 3:13-19, and Luke 6:12-19
 The Twelve
 Psa 41:9; Psa 69:4, 25
This is an on going study and is subject to revisions.