The genealogy in Matthew 1:1–17 is not intended to be complete. Instead, he uses it to summarize the history of Israel, in three different periods: Abraham to David, David to the deportation to Babylon, and the deportation to Christ. This type of summary through a genealogy had been done earlier in Genesis 5:1–32, Genesis 10:1–32, Genesis 11:1–32, and Genesis 25:1–34, and in Ruth 4:18–22. Matthew shows how the line of Abraham reached its highest point in the line of David. His summary traces God’s messianic promise back to Abraham, and the promise of a King back to David.
No, Matthew’s focus is specifically on the history of Israel, which he strives to summarize by showing how the line with the patriarchs Abraham, Isaac, and Jacob culminated in the royal house of David (see Matthew 1:17).1
Matthew ultimately wants to provide a survey of a period of history between Abraham and Jesus. Writing history in the form of a panorama of generations is known from the book of Genesis (Genesis 5:1–32, Genesis 10:1–32, Genesis 11:1–32, Genesis 25:1–34; see also Ruth 4:18–22).2
We find the main thesis in Matthew 1:17. There Matthew divides the history into three equal parts: from Abraham to David, from David to the deportation to Babylon, and from the deportation to Babylon to Christ. The first two were already fixed in Matthew 1:1b (Abraham and David). Those are not remarkable. The special history of the promise began with Abraham, while the special history of the kings began with David.3
2 Abraham was the father of Isaac, and Isaac the father of Jacob, and Jacob the father of Judah and his brothers,