Και αυτος ἦν ὁ Ιησους, ὡσει ετων τριακοντα αρχομενος, ων ὡς ενομιζετο υἱος Ιωσηφ, του Ηλει, του Ματθαν, του Λευι, του Μελχι…

There are four words here that invite special attention: και αυτος ἦν and ὡς. Since verse 22 ends with a statement from the Father at Jesus' baptism, it is clear that verse 23 begins another section. But the conjunction that signals the transition is και and not δε, as one would expect—this means that ‘Jesus’ continues as the topic. But in that event, how does one explain the personal pronoun αυτος, the more so in such an emphatic position? If the author's purpose was simply to register Jesus as a son of Joseph, as many suppose, why didn't he just write και ὁ Ιησους ἦν υἱος Ιωσηφ, etc.?

But then, why write ὡς ενομιζετο? It seems to me that the normal meaning of “as was supposed” is to affirm that Jesus was in fact Joseph's son; but that is precisely what Jesus was not. Luke has already made clear that Jesus' real Father was the Holy Spirit—1:34-35, 43, 45; 2:49. So what Luke is really saying is that although the people supposed Jesus to be Joseph's son, He actually had a different lineage—we should translate “so it was supposed”. (Recall that a faithful and loyal translation seeks to transmit correctly the meaning intended by the author.)

The verb ἦν is the only independent one in the whole paragraph, verses 23-38. Is it working with the participle αρχομενος in a periphrastic construction? That appears to be the tendency of the eclectic text that places the participle right after Jesus (following less than 2% of the Greek MSS), which makes Jesus out to be in fact Joseph's son. It seems to me to be far more natural to take the participial clauses as being circumstantial: “beginning at about thirty years of age” and “being (so it was supposed) a son of Joseph”. Setting those two clauses aside, the independent clause that remains is ἦν ὁ Ιησους του Ηλει, “Jesus was of Eli”.

The participle ‘beginning’ requires an object, that the Text leaves implicit; from the context it seems clear that we may supply ‘His ministry’, or some such thing, which is why most versions do so. I suggest the following rendering: “Beginning His ministry at about thirty years of age, being (so it was supposed) a son of Joseph, Jesus was actually of Eli, of Mathan, of Levi…” I take it that the emphatic pronoun αυτος heightens the contrast between what the people imagined and the reality. Jesus was a grandson of Eli, Mary's father—Luke gives the genealogy of Jesus through His mother, while Matthew gives it through His stepfather. Jesus received some of David's genes through Mary and Nathan; the glorified body now at the Father's right hand, and that will one day occupy David's throne, has some of his genes.

The eclectic text gives our verse a different wording: και αυτος ἦν Ιησους αρχομενος ὡσει ετων τριακοντα, ων υἱος, ὡς ενομιζετο, Ιωσηφ του Ηλι του Μαθθατ του Λευι του Μελχι… The RSV translates it like this: “Jesus, when he began his ministry, was about thirty years of age, being the son (as was supposed) of Joseph, the son of Heli, the son of Matthat…” Is not the normal meaning of this rendering that Jesus was in fact the son of Joseph? However, every version that I recall seeing has “Joseph, the son of Heli”, which directly contradicts Matthew, “Jacob begot Joseph”. The word ‘son’ (without the article) occurs only with Joseph, although most versions supply it on down the genealogy. But Luke is precisely correct in not using it, because it would not hold for the first and last names in the list—Eli did not beget Jesus (nor Joseph) and God did not beget Adam.

So then, properly understood Luke does not contradict Matthew (with reference to Joseph's father), nor does he affirm an error of fact (with reference to Jesus' father).