Unbelievers in the New Covenant?

According to Romans 9:6 "... not all who are descended from Israel are Israel." The question arises: Who does Paul mean by the first "Israel" and who does Paul mean by the second "Israel"? It is evident from the passage that "all who are descended from Israel" are all of Jacob's physical descendants or as Romans 9:8 puts it "the natural children". So then the verse becomes "... not all who are the physical descendants of Jacob are Israel." Romans 9:8 contrasts "natural children" with "children of the promise" but equates "God's children" with "Abraham's offspring". Since we have already stated that "natural children" is the same as the physical descendants of Jacob, we can say that the second Israel of Romans 9:6 are God's children and Abraham's offspring [both Jews and Gentiles].

According to Galatians 3:29, "If you belong to Christ, then you are Abraham's seed, and heirs according to the promise." Here we see that "seed" means "Christian". Similarly, John 1:12,13 teaches that "to those who believed in his name, he [Christ] gave the right to become children of God — children born not of natural descent, nor of human decision or a husband's will, but born of God." Note again the contrast of "natural descent" with "children of God" and "born of God".

It is clear, then, that the second Israel of Romans 9:6 means those who belong to Christ, that is, the church [from the time of Adam forward - Jews and Gentiles]. It is also evident that many in this passage who part of the covenant community in Israel and, by extension) the Church are not necessarily children opf promise.
-Andrew Wheatley (slightly modified)