Why are the women in Matthew's genealogy included?

There are ongoing disagreements about why Matthew refers to Tamar, Rahab, Ruth, and Bathsheba in Matthew 1:3-6. It's sometimes suggested that they're included because of the theme of God's acceptance of Gentiles. That theme is prominent in early Christianity, including in the gospel of Matthew, as the magi in chapter 2 illustrate. But Bathsheba apparently was Jewish. She lived in Isarael. In 2 Samuel 11:3, her father is referred to as Eliam, with no further qualifier, whereas the same verse qualifies Uriah as "the Hittite". See, also, 2 Samuel 23:34. If Ahithophel was the grandfather of Bathsheba, which would be further evidence of her Jewishness, that would help explain why Ahithophel betrayed David (because of what David did to Bathsheba). Matthew 1:16 includes Mary, who was Jewish, in a way similar to how the other women are included earlier. Furthermore, when Bathsheba is referred to in verse 6, she's identified as the wife of Uriah, not "the wife of a Hittite", "the wife of Uriah the Hittite", or some other such thing. And she's referred to as Uriah's wife in a context about giving birth to Solomon after marrying David. She wasn't Uriah's wife at the time, yet Matthew chose to mention that she had been Uriah's wife. The focus seems to be on the adulterous origins of her relationship with David, not any connection to Gentiles. Matthew probably didn't think Ruth was guilty of sexual sin, and he didn't think Mary was, so he didn't think the women had sexual sin in common. Even the women who were sexually immoral were so in significantly different ways. For example, Rahab's background as a prostitute is substantially different than conceiving the child mentioned in the genealogy by means of sexual immorality. It also seems unlikely that the women were all thought to have had a low social status independent of issues like sexual immorality (being born into a disreputable family, being of low economic status, etc.). As explained above, Matthew highlight's Bathsheba's involvement in adultery, which is distinct from the sort of social status issues I just referred to. What's the common thread with these women, then?

Ruth 4 And Christmas

The last chapter of Ruth should get more attention than it does during the Christmas season. Notice the references to Gentile inclusion, building the house of Israel, fame related to Bethlehem, a supernatural conception, and Davidic ancestry, for example. Much of what's in the passage is somewhat reminiscent of Christmas themes, and some of what's anticipated has been fulfilled more by Jesus than by David:

"'I have acquired Ruth the Moabitess [a Gentile]'…'May the Lord make the woman who is coming into your home like Rachel and Leah, both of whom built the house of Israel; and may you achieve wealth in Ephrathah and become famous in Bethlehem.'…So Boaz took Ruth, and she became his wife, and he went in to her. And the Lord enabled her to conceive, and she gave birth to a son. Then the women said to Naomi, 'Blessed is the Lord who has not left you without a redeemer'…to Boaz [was born], Obed, and to Obed was born Jesse, and to Jesse, David." (Ruth 4:10-11, 4:13-14, 4:21-22)

"The stories of Jesus's birth are the foundation of the world's most widely observed holiday. Christmas is celebrated by the world's two billion Christians, a number about twice that of the next largest religion, Islam. Moreover, because of the cultural and commercial importance of Christmas in Western culture and beyond, it is observed by many non-Christians as well. Indeed, no other religious holiday is so widely commemorated by people who are outside of the tradition that originated it....Since Matthew and Luke agree independently on those two points about Jesus - that he was descended from David's lineage and born in David's city [Bethlehem] - those must come from an earlier tradition than either of their Christmas stories. And, in fact, we find both of those points elsewhere in the New Testament." (Marcus Borg and John Dominic Crossan, The First Christmas [New York, New York: HarperCollins Publishers, 2007], vii, 130)