Comparison between Matthew 1:16 and Luke 3:23
ιακωβ δε εγεννησεν τον ιωσηφ τον ανδρα μαριας εξ ης εγεννηθη ιησους ο λεγομενος χριστος
iakób de egennésen ton ióséph ton andra marias ex és egennéthé iésous o legomenos kristos
Jacob begot Joseph, the husband of Mary, from whom was born Jesus, who is called Christ.
y Jacob engendró a José, el marido de María, de la cual nació Jesús, quién es llamado Cristo.
και αυτος ην ιησους αρχομενος ωσει ετων τριακοντα αρχομενος ων υιος ως ενομιζετο υιος ιωσηφ του ηλι
kai autos én iésous arkomenos ósei etón triakonta ón uios ós enomizeto ióséph tou éli
And he was Jesus, when he began, about thirty years old, being son (as was thought) of Joseph of Heli
Y él tenía Jesús, [cuando] comenzó, unos años treinta, siendo hijo (como se suponía) de José de Elí
And when Jesus began, he was about thirty years old, being the son (as was thought) of Joseph of Heli
Y cuando comenzó Jesús, él tenía unos treinta años, siendo el hijo (como se suponía) de José de Elí
Matthew 1:16 says that Jacob begot Joseph. The word begot (εγεννησεν) clearly implies that Jacob was the biological father of Joseph.
Luke 3:23 does not use the word begot at all when linking Heli and Joseph in this geneaology. The phrase is literally, "Joseph of Heli". The Greek word "tou" (του), simply means "of the"--in this case "of Heli". This does not state at all that Joseph is here said to be the biological son of Heli, as was the case with Jacob Matthew 1:16. Therefore no contradiction can be dogmatically implied here without first examining the rest of the available evidence in favor of the Joseph-Mary geneaology theory.
The difference between the two different words that link Joseph with two different fathers is extremely important to notice, because of the fact that there are two very different geneaologies recorded.
The theory that Joseph is the biological son of Jacob, but the son-in-law of Heli, who was Mary's father, could be evidenced by the Jewish Hagigah 77,4. This non-Christian text can at least be referenced in showing that a son-in-law was referred to as a son, and Mary was described as the daughter of Heli.
The problem with these two parallel passages is that people think they contradict one another when they do nothing of the sort. They are demonstrably two different family trees. One of Joseph's family and the other of Mary's family.
Matthew's is the family line of Joseph, while Luke's is the family line of Joseph's father-in-law, Heli, thus making it Mary's family line. What this shows the reader is that both Joseph and Mary were in the line of David. The reason that is so important is because the Messiah was prophesied to be a direct descendant of David. Joseph was a descendant of David, as Matthew showed in his geneaology. But the Bible tells us that Joseph was not the biological father of Jesus, since Jesus was virgin born. Therefore Luke shows how Mary was biologically from the family line of David, too. So we can see from these two different geneaologies that the promise from God that His Servant, the Christ, would be from the family of David was fulfilled in both Joseph and Mary.
Another important factor to consider is that family geneaologies always used the man's name in that culture and time period, even as it were, when it was talking about the woman's family. It can also be demonstrated, both historically and Biblically, that a father-in-law could refer to his son-in-law simply as a son, due to the legal marriage with his daughter. This was another common practice of the Jews.
So the difference in the words used for the two fathers of Joseph, the two different geneaologies recorded, the demonstration of the Messianic promise fulfilled in both Joseph and Mary as the parents of Jesus, as well as the historic and Biblical practice of the Jews in using the man's name in geneaological records all points to the fact that these differences are not even close to contradictions, but are two intentional and informative accounts of the family line of Jesus Christ.
With these demonstrable facts in mind, we ask the reader to reconsider whether those who claim this is a contradiction are not the ones truly making an error in their judgments of these passages.
(Note: there are other questions and responses to other parts of these geneaologies, but for now we have simply tried to briefly answer the question about the two different fathers of Joseph and the identities of the two geneaologies)