Melchizedek is human, not greater than Christ, and his missing genealogy is a type
Chrysostom is reporting the orthodox reply to those who said Melchizedek was greater than Christ or was the Holy Spirit, and to Jews who said he was born of a prostitute.
But we say that he was a human being with feelings like ours, not greater than Christ—nor even greater than John the Baptist—and certainly not the Holy Spirit. As for the claim that he is greater than Christ, let those who say this tell us: what region do they say he belongs to? Is it heavenly? Is it earthly or under the earth? If it is heavenly or some other region, let them hear that to him every knee will bend, and so on (Philippians 2:10). If every knee bends, then the one who worships would be less than the Christ who is worshipped. But the wretched people do not even know the next line: “being made like the Son of God”—just as we also came into being according to the image and likeness of God (Genesis 1:26).
But perhaps one of the hearers asks and says: What then does it mean that he is called without father, without mother, and without genealogy? To him we will answer that the Jews have said he was born of a prostitute, and for that reason Scripture did not trace his genealogy. To them we will answer: You have spoken wrongly. For we find others who were born of a prostitute and yet had their genealogy traced—immediately Solomon, born of Uriah’s wife, and then Abimelech, the son of Gideon. Why then is his genealogy not traced? Because he was a type of Christ and bore his image. For this reason Scripture omitted his father and mother, so that in him, as in an image, we might behold as in a mirror the Christ who is truly without father, without mother, and without genealogy.
Greek
Ἡμεῖς δὲ λέγομεν αὐτὸν
εἶναι ἄνθρωπον ὁμοιοπαθῆ ἡμῖν καὶ μήτε μείζονα Χριστοῦ, ἀλλ' οὔτε Ἰωάννου τοῦ βαπτιστοῦ, οὔτε μὴν εἶναι αὐτὸν τὸ Πνεῦμα τὸ ἅγιον. Ὅτι δὲ οὐκ ἔστι μείζων τοῦ Χριστοῦ, εἰπάτωσαν ἡμῖν οἱ τοῦτο λέγοντες ποίου χωρίου αὐτὸν εἶναι λέγουσιν; ἆρα οὐρανίου; ἆρα ἐπιγείου ἢ καταχθονίου; Εἰ μὲν οὐρανίου ἢ ἄλλου τινὸς χωρίου, ἀκουσάτωσαν ὅτι αὐτῷ κάμψει πᾶν γόνυ καὶ τὰ ἑξῆς. Εἰ οὖν κάμπτει γόνυ, ἐλάττων εἴη ὁ προσκυνῶν τοῦ προσκυνουμένου Χριστοῦ. Ἀλλ' οὔτε τὸν ἐπιόντα στίχον ἴσασιν οἱ τάλανες· τῷ ἀφομοιουμένῳ τῷ υἱῷ τοῦ Θεοῦ· ὥσπερ καὶ ἡμεῖς κατ' εἰκόνα καὶ ὁμοίωσιν Θεοῦ ἐγενόμεθα. Ἀλλ' ἴσως τις ἐπερωτᾷ τῶν ἀκροατῶν καὶ λέγει· τί οὖν ἔστι τὸ ἀπάτορα αὐτὸν λέγεσθαι καὶ ἀμήτορα καὶ ἀγενεαλόγητον; Πρὸς ὃν ἐροῦμεν ὅτι Ἰουδαῖοι μὲν αὐτὸν εἰρήκασιν ἐκ πόρνης γεννηθῆναι καὶ διὰ τοῦτο μὴ γενεαλογηθῆναι ὑπὸ τῆς γραφῆς. Πρὸς οὓς ἐροῦμεν ὅτι κακῶς εἴπατε. Εὑρίσκομεν γὰρ καὶ ἄλλους ἐκ πόρνης γεννηθέντας καὶ γενεαλογηθέντας· εὐθὺς ὁ Σολομῶν ἐκ τῆς τοῦ Οὐρίου, εἶτα καὶ ὁ Ἀβιμέλεχ υἱὸς Γεδεών. Τίνος οὖν ἕνεκεν οὐ γενεαλογεῖται; Ἐπειδὴ τύπος ἦν τοῦ Χριστοῦ καὶ τούτου εἰκόνα ἔφερε, διὰ τοῦτο παρέλιπεν αὐτοῦ ἡ γραφὴ τὸν πατέρα καὶ μητέρα, ἵνα ἐν ἐκείνῳ καθάπερ εἰκόνι τὸν ἀληθῶς ἀπάτορα καὶ ἀμήτορα καὶ ἀγενεαλόγητον Χριστὸν ἐνοπτρισώμεθα.