In Galatians 4, it seems that Paul intentionally used γενέσθαι when speaking of Jesus while γεννηθῆναι when talking of Isaac and Ishmael being born of Sarah and Hagar, respectively.
Here are the verses in question:
4:4-5 ὅτε δὲ ἦλθεν τὸ πλήρωμα τοῦ χρόνου, ἐξαπέστειλεν ὁ θεὸς τὸν υἱὸν αὐτοῦ, γενόμενον ἐκ γυναικός, γενόμενον ὑπὸ νόμον, ἵνα τοὺς ὑπὸ νόμον ἐξαγοράσῃ, ἵνα τὴν υἱοθεσίαν ἀπολάβωμεν.
4:23 ἀλλ’ ὁ μὲν ἐκ τῆς παιδίσκης κατὰ σάρκα γεγέννηται, ὁ δὲ ἐκ τῆς ἐλευθέρας δι’ ἐπαγγελίας.
The epistles of John also use γεννηθῆναι when they are talking about believers being "born of God."4:29 ἀλλ’ ὥσπερ τότε ὁ κατὰ σάρκα γεννηθεὶς ἐδίωκε τὸν κατὰ πνεῦμα, οὕτως καὶ νῦν.
There's obviously a distinction to be made here. Richard Carrier makes the argument that γίνομαι (i.e., γενόμενος) refers to Jesus' being "made" (similar to ποιησάμενος) rather than "born," since Paul didn't envision Jesus as a human being in the same sense as we are human beings. That is, he envisioned him as an angelic being that took on flesh in some sense (by descending through the various heavens rather than actually having been born to a human mother).
Can you devise any explanation that might clarify why Paul used γίνομαι of Jesus rather than γεννηθῆναι? That is, why didn't he write γεννηθέντα ἐκ γυναικός instead of γενόμενον ἐκ γυναικός? I'm wondering how much explanatory power Carrier's argument has. It attracts me, and I can certainly see that Paul distinguishes here between these two verbs (on purpose, for sure). I'm just looking to find out if there is another (better?) explanation for this lexical distinction.
εὐχαριστῶ τοῖς ἀποκρινομένοις πᾶσιν ἐν ἀγαπήσει καὶ φιλίᾳ.
Ἰάσων τοῦ Ἰωάννου