Let's focus on Abraham's first two sons. The older was born through a bondwoman, Hagar. His name was Ishmael. The younger was born to Sarah, Abraham's wife. His name was Isaac.
Ishmael, the older was conceived and born the old-fashioned way, through the flesh.
Isaac, on the other hand, was a child of promise, enabled by the Spirit of God.
There's an allegory in the family dynamic. The status of wives, nature of birth, and ultimate relationship to Abraham were all played out in history. But, as is so common in OT history, the facts of history were prophetic. Each of the five people in Paul's short flash-back served to allegorize the Divine redemption story.
Abraham stood for the Father. Sarah stood for the Bride. Hagar represented the slave-wife (to soon be divorced--Paul's "soon"). Isaac pre-figured the sons of the promise, who were to inherit from the Father. And Ishmael depicted OT Israel, who were locked as slaves in their sins, though loved by their Father.
Only the free, promise-born sons were set to inherit. The residue were about to be cast out. Do not revert to the slave-son status, Paul argues. Do not put yourselves in the "about to be cast out" group.
Join the study; see the argument unfold.