Galatians 3:7

‘Know therefore that those who are of faith, those are the sons of Abraham.'

Thus the true sons of Abraham are those who like him exercise faith in the promises of God and are accounted righteous accordingly (John 8:39), and they then live accordingly. But the way that they live is the result of their salvation, not the cause of it. The cause of it is the work of the Spirit within through the gracious mercy and favour of God. They walk in faith, they walk in the Spirit. Compare also Matthew 3:9; Luke 3:8 where John the Baptiser says that those who claim to be sons of Abraham in the flesh are not superior to anyone else, simply because God is able to raise sons to Abraham from the very stones around them. Being a son of Abraham was nothing special. It was having the faith of Abraham that was special. Whatever claims the Judaisers made, then, they were not behaving like sons of Abraham, for they looked to themselves and their own works to establish their righteousness, rather than to God in faith, while Abraham did nothing except believe, and simply looked to God, and receive from Him whatever He wanted to give him.