The New Covenant
The New Covenant is first explained by the prophet Jeremiah, who wrote in Jer. 31:31-33,
31 Behold, days are coming, declares the Lord, when I will make a new covenant with the house of Israel and with the house of Judah, 32 not like the covenant which I made with their fathers… 33 But this is the covenant which I will make . . . I will put My law WITHIN them, and on their heart I will write it; and I will be their God, and they shall be My people.
There are two main differences set forth here between the two covenants. First, the New Covenant is an unconditional covenant, because there is no "IF" clause. It is a covenant where God promises to do the work in us, rather than making the promise conditional upon man's works or efforts.
Secondly, the New Covenant is based upon an inner work that God will do, because the law will no longer be written externally on tables of stone, but internally upon the heart. instead of the law being imposed by man—either by other men or by our own wills through self-discipline—it is accomplished through the Holy Spirit. He speaks to us and leads us by His Spirit, even as Israel was led by the pillar of cloud and the pillar of fire in the wilderness under Moses.
These two covenants, then, are allegorically pictured in Ishmael and Isaac. The children of the Old Covenant are enslaved, attempting to become perfected by their own ability to be perfectly obedient. Their sincerity cannot be questioned and can even be very admirable. Many are able to achieve great powers of the mind and can even walk on hot coals without burning their feet. But the bottom line is that they must surely fail to fulfill the legal requirements of a righteous God.
On the other hand, the children of the New Covenant are born of a freewoman, even as Isaac. The responsibility is placed upon God's shoulders to make His people righteous. This was done through the fulfillment of the feast of Passover when Jesus Christ paid the penalty for our sin through His death on the cross. And now, as we journey through the wilderness of life, we have a second feast called Pentecost in which God is writing His law upon our hearts.
Christians are those whose relationship with God is defined by the provisions of the New Covenant. When they accept Jesus Christ's Sacrifice for sin, God imputes the righteousness of Christ to them--as if they were already perfected. They are then able to journey through the wilderness (Pentecost) as freemen, rather than as bondmen. They are IMPUTED righteous until such time as they are made ACTUALLY righteous at the fulfillment of the feast of Tabernacles.
Thus, the path to Sonship is in three steps defined by these three feast days.
Another difference between the Old and New Covenants is that each is associated with a particular SIGN. The sign of the Old Covenant is fleshly circumcision, as instituted first by Abraham in Gen. 17:11. The sign of the New Covenant is the circumcision of the heart.
This heart circumcision was first mentioned as far back as Moses, who said in Deut. 30:6,
6 Moreover the Lord your God will circumcise your heart and the heart of your descendants, to love the Lord your God will all your heart and with all your soul, in order that you may live [i.e., have immortality].
Moses viewed this heart circumcision as being something that God Himself would have to do in us. No man can circumcise his own heart. For this reason, it is associated with the New Covenant, because it is something that God does in us, rather than something that man does by his own works or self-discipline.
And so we see that the Bible sets forth Ishmael and Isaac as allegories of two covenants, two different ways of attaining righteousness, with two different inheritances. Those who follow the path of Ishmael—regardless of who they are or what race they are--are of the Old Covenant. Those who follow the path of Isaac are of the New Covenant.
Although this is not a racial statement as such, history shows that the physical descendants of Ishmael have come into a religion called Islam, which by its very name signifies obedience and thus perfectly describes the Old Covenant.
But in Galatians 4, Paul also identifies the earthly city of Jerusalem with Hagar, and her "children" with the adherents of Judaism. Judaism has bound itself to the Old Covenant and refuses to accept the New Covenant. Thus, in that legal sense, the Jews are "Ishmael," because they consider their "mother" to be Jerusalem (Hagar).
The Old Covenant is incapable of bringing forth the sons of God, and so neither Islam nor Judaism is able to obtain the inheritance of the sons of God. Their inheritances are earthly and have more to do with physical land (real estate), rather than the glorified body, which is the greatest "land" inheritance.