Ignatius wrote in A.D. 110 that Jesus was gennetos kai agennetos, meaning “begotten and not begotten.”
Did Ignatius believe in the doctrine of Eternal Generation of the Son? Why did he say that Jesus is not begotten in his divinity?
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This "begotten and unbegotten" quote comes from Ignatius's Epistle to the Ephesians, chapter 7.2. One source for the Greek is The Loeb Classical Library, Apostolic Fathers, vol. 1, page 180. English translations of the text differ somewhat, but here are two popular ones:
There is one Physician who is possessed both of flesh and spirit; both made and not made; God existing in flesh; true life in death; both of Mary and of God; first possible and then impossible,—even Jesus Christ our Lord. (ANF)
There is one Physician, who is both flesh and spirit, born and yet not born, who is God in man, true life in death, both of Mary and of God, first passible and then impassible, Jesus Christ our Lord. (Loeb, 181)
Quoting this section, J. N. D. Kelly, in Early Christian Doctrines, 92, indicates that the word agennetos is "the technical term reserved to distinguish the increate God from creatures." Thus the emphasis can be understood to be more on Christ's distinction from his creatures than between Him and the Father.
Still, this is an attribute typically associated with God the Father, and this isn't the only place where Ignatius closely connects the Son and Father – Kelly provides several other quotes along similar lines, and describes how this language has sometimes been interpreted:
In view of this language the conclusion has sometimes been drawn that [...] Ignatius was really an 'economic trinitarian', i.e. regarded God as an undifferentiated monad in His essential being, the Son and the Spirit being merely forms or modes of the Father's self-revelation, only distinguishable from Him in the process of revelation. (93)
However, Kelly disagrees with this conclusion, and contends that Ignatius did believe in the pre-existence of the Son – not that he only became an expression of God in the incarnation – as well as his independence from the Father:
[The thought of Ignatius] was steeped in the Fourth Gospel, and its strong emphasis on the oneness of Christ with the Father reflects such Johannine texts as 1:1f.; 10:30; 14:9; 17:5. [...]
[Ignatius] definitely states that He 'existed with the Father before the ages', and that He 'came forth from the unique Father, was with Him and has returned to Him'. Phrases like these imply a real distinction, as do the passages in which he compares the relation of deacons to the bishop, or of the church to the bishop, to that of Christ to the Father.
Numerous other contexts suggest that His independence vis-a-vis the Father was not limited to His earthly sojourn, such as (a) the formulae of greeting and farewell affixed to the letters, and (b) Ignatius's requests to his correspondents to address their prayers to Jesus Christ. (93)
So it's fair to conclude that while Ignatius did closely associate the Father and the Son, this use of "unbegotten" can be understood as referring to Christ as compared to his creatures, and that Ignatius did believe in the eternal generation of the Son.
It seems what you quote is a Trinitarian formula of his as below:
Chapter 14. Conclusion | Epistle to the Antiochians | Spurious Epistles of St. Ignatius of Antioch | New Advent
I write this letter to you from Philippi. May He who is alone unbegotten, keep you steadfast both in the spirit and in the flesh, through Him who was begotten before time began! And may I behold you in the kingdom of Christ! I salute him who is to bear rule over you in my stead: may I have joy of him in the Lord! Fare well in God, and in Christ, being enlightened by the Holy Spirit.
It is clear that the Unbegotten is the Father.
In all his documents, Jesus, the Son, and the Trinity, are as per the Apostolic and Catholic Teaching.
From, Excursus on the Words Gennethenta Ou Poiethenta., my understanding is that Jesus as man is generated in time, his Divinity was never generated when he appeared as God-made-man, and he was begotten of the Father before time began.
cf. Athanasian Creed.
About St. Ignatius of Antioch
Born is Syria died in Rome between 98 and 117.
May have been the child whom Our LORD took in his arms (cf. Mk 9:36)
His friend was St. Polycarp, a disciple of St. John the Apostle.
Was the third bishop of Antioch appointed by St. Peter himself.
Source: Glimpses of the Church Fathers | Claire Rusell