"These things may be taken figuratively, for the women represent two covenants. One covenant is from Mount Sinai and bears children who are to be slaves: This is Hagar. Now Hagar stands for Mount Sinai in Arabia and corresponds to the present city of Jerusalem, because she is in slavery with her children. But the Jerusalem that is above is free, and she is our mother." NIV translation
Mount Sinai represents the Mosaic Law and, as we saw in chapter three, the law does not have the power to free us. Therefore, those who trust in the law remain slaves and this corresponds to Hagar and her son Ishmael. Today, the equivalent are those who practice Islam, which began in Arabia, as they are slaves to rituals that cannot bring salvation and freedom. Likewise, the city of Jerusalem today is captive to those who are trying to earn their freedom. But, the new Jerusalem which is coming (see Revelation 21) is not for those who try to earn it but those that accept the freedom that comes through faith in Jesus Christ.
"For it is written: 'Be glad, O barren woman, who bears no children; break forth and cry aloud, you who have no labor pains; because more are the children of the desolate woman than of her who has a husband.' Now you, brothers, like Isaac, are children of promise." NIV translation
Paul quotes from the prophet Isaiah (54:1), in order for us to see that we are children of the same promise that was given to Abraham and Sarah. The quote is a call for the relatively small nation of Israel (God's people) to rejoice in the fact that there was more of God's people than what they were seeing. Although they could not see it at the time, God was looking down through time and speaking about the adoption of Gentiles through the blood of Jesus Christ (a Jew).
"At that time the son born in the ordinary way persecuted the son born by the power of the Spirit. It is the same now." NIV translation
After Isaac was born, Ishmael mocked him by trying to claim his promise. It is the same today as we see the battle between Islam (Ishmael) & Judaism (Isaac). This battle is displayed in attempts to claim Jerusalem and parts of Israel as well as the shedding of Jewish and Christian blood.
"But what does the Scripture say? 'Get rid of the slave woman and her son, for the slave woman's son will never share in the inheritance with the free woman's son.' Therefore, brothers, we are not children of the slave woman, but of the free woman." NIV translation
Paul quotes from Genesis 21 and this is where God tells Abraham these words. Those who are identified with Ishmael and his descendants do not share in the inheritance which is salvation through Jesus Christ. Jesus was a descendant of Isaac and so all who are his are also descendants of Abraham through Isaac. Therefore, we are children of the promise and share in the inheritance through Jesus. We are, in fact, sons of God.