In this age, the community of the messiah exists in concentric circles. The outer circle (circulus ad extra) is porous and visible to the world. The inner circle (circulus ad intra) is nonporous and impenetrable. The outer circle is the visible people of the messiah "built on the foundation of the apostles and prophets," the Old Testament Israel and the New Testament church. The inner circle is the elect "chosen in him before the foundations of the world." And, as in Jesus's parable of the wheat and the tares (Matt. 13.24-30), "both of them grow together until the harvest."
It is easy to enter the outer circle. No one can forbid it. One is born into it or one bids for membership out of personal desire or in response to an invitation. And there is a liturgical sign of entry. In Deuteronomy 28:9, directly after God says "The Lord will establish you as his holy people," he commands circumcision as a visible token of membership. Male babies were to be circumcised shortly after birth, and through them, the nation is circumcised. Circumcision was replaced by baptism in the Messianic, New Testament community. Members of the circle have public access to the benefits of the community.
It is impossible, by human effort, to enter into the inner circle (see Opening a Closed Circle, 83). Its members are elect by God. The language and benefits of the community are reserved for and active in the elect alone. They are the invisible community of the inner circle "born, not of blood nor of the will of the flesh nor of the will of man, but of God." The inner circle is the church militant (Ecclesia militans). It is the burning bush in which God dwells by the Spirit and every man, woman, and child is on fire.
It is impossible and evil to fix another's place on the inner or outer circle, aside from their making an overt declaration. And it can be challenging, in the day to day, to know where to place oneself.
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[1] The history of the dogma of an invisible and invisible church begins with Martin Luther. Luther's protest against Roman Catholicism demanded a new answer to the question "Who makes up the church?"
Luther called the Roman Catholic Church a “false church” (his sermon Von der heiligen christlichen Kirche in 1531). He set a true or hidden church over against it. His hidden church was separate from the visible church, the workaday church or the communio sanctorum of human beings harassed by sin and suffering, existing on and by God’s grace. This visible church was a mixture of good and bad. It was streaked with sin and harbored sinners even though it was diligent to combat sin and the perversion of doctrine. Inside it was the hidden church. "The church must appear in the world. But it can only appear in a covering (larva), a veil, a shell, or some other kind of clothes which a man can grasp, otherwise it can never be found," (WA Briefwechsel 9, 608). For Luther, then, the church is like a sacrament: the visible and hidden can be distinguished but not separated in this age. The visible may be handled and, underneath it, the hidden made available.
The false church does not value the scriptures. And, therefore, it cannot properly offer the sacraments or preach the gospel. The false church has no life in it--no inner grace, no hidden church.
Luther also perceived an analogy to his visible-and-hidden ecclesiology in the incarnation of Jesus, where God and man are one. The hidden church suffers in its flesh, even as its head, Jesus, suffered. And through that suffering the gospel life is made available to human beings.
Calvin, as is to be expected, sums up the work of the first Reformers:
"The Church," he says, "is used in the sacred Scriptures in two senses. Sometimes when they mention the church, they intend that which is really such in the sight of God (quae revera est coram Deo) into which none are received but those who by adoption and grace are the children of God and by the sanctification of the Spirit are the true members of Christ. And then it comprehends not only the saints at any one time resident on earth but all the elect who have lived from the beginning of the world.
"But the word church is frequently used in the Scriptures to designate the whole multitude dispersed all over the world who profess to worship one God and Jesus Christ, who are initiated into his faith by baptism, who testify their unity in true doctrine and charity by a participation of the sacred supper, who consent to the word of the Lord and preserve the ministry which Christ has instituted for the purpose of preaching it. In this church are included many hypocrites, who have nothing of Christ but the name and appearance; many persons, ambitious, avaricious, envious, slanderous, and dissolute in their lives, who are tolerated for a time, either because they cannot be convicted by a legitimate process, or because discipline is not always maintained with sufficient vigor.
"As it is necessary therefore to believe that Church which is invisible to us, and known to God alone, so this Church, which is visible to men, we are commanded to honor, and to maintain communion with it." (Institutes Bk. IV.1 § 7)
This post is a work in progress. Its existence is to be a short, model ecclesiology, combining my model of "Saint Augustine's church", something on the two sacraments, and election. Note: your grammar is not quite right. Barth has some good tweaks. see https://postbarthian.com/2016/04/29/ karl-barths-rejection-visible-invisible-church/ As if to say that the circle one should care about is not the circle of exclusion between people, but the circle of election. Previous post links an available hermeneutic to a welcoming ecclesiology. The mission-stance of the church mirrors the mission-stance of the Triune God. You were not wrong to bring in the Trinity ad extra / ad intra. Put that back in. The private ascesis of the Sermon on the Mount can be appended to this as well. The circle of the Spirit is wider than the circle of the church. All people are made in the imago Dei, albeit now marred by sin. Only saints are made into a kingdom of priests
