In Romans chap. 9 Paul makes some interesting statements where he contrasts Israel of the flesh and Israel of the promise or spirit:
I say the truth in Christ, I lie not, my conscience also bearing me witness in the Holy Ghost, That I have great heaviness and continual sorrow in my heart. For I could wish that myself were accursed from Christ for my brethren, my kinsmen according to the flesh: Who are Israelites; to whom pertaineth the adoption, and the glory, and the covenants, and the giving of the law, and the service of God, and the promises; Whose are the fathers, and of whom as concerning the flesh Christ came, who is over all, God blessed for ever. Amen. Rom. 9:1-5
He then goes on to say that the children of the flesh are not the children of God or merely being born into the nation of Israel did not guarantee being part of the family of God.
Not as though the word of God hath taken none effect. For they are not all Israel, which are of Israel: Neither, because they are the seed of Abraham, are they all children: but, In Isaac shall thy seed be called. That is, They which are the children of the flesh, these are not the children of God: but the children of the promise are counted for the seed. Rom. 9:6-8
The children of the promise are counted as the seed or offspring of Abraham. The original promise to Abraham was that God would make him a great nation and then he delayed that promise to test Abraham's faith. There were two nations that sprang from Abraham and they were not Isaac and Ishmael. Both of these men fathered fleshly nations. The nation of the promise are all those who have the same faith that Abraham did and Abraham's faith was severely tested. A mere profession is worthless just like trusting in genealogies is worthless.
As he saith also in Osee, I will call them my people, which were not my people; and her beloved, which was not beloved. And it shall come to pass, that in the place where it was said unto them, Ye are not my people; there shall they be called the children of the living God. Rom. 9:25,26
What shall we say then? That the Gentiles, which followed not after righteousness, have attained to righteousness, even the righteousness which is of faith. But Israel, which followed after the law of righteousness, hath not attained to the law of righteousness. Wherefore? Because they sought it not by faith, but as it were by the works of the law. For they stumbled at that stumblingstone; As it is written, Behold, I lay in Sion a stumblingstone and rock of offence: and whosoever believeth on him shall not be ashamed. Rom. 9:30-33