For example, I grew up in a church and denomination that had a clear definition of membership. It is a minimalist definition, but a definition nonetheless. To be a member of an ELCA congregation, you have to commune in the last calendar year, and give "of note," meaning in an offering envelope or connected to your name in some way.
Many have complained that this isn't enough, it doesn't really signify true membership. One Sunday at communion, and a penny in an envelope, does probably not a disciple make.
However, such definitions serve a good purpose inasmuch as they set up some healthy boundaries that prepare congregations to address complex issues like who can vote at the annual meeting, and who can serve on council. A random group who has never given and never worshipped in a church can't just show up at the annual meeting and claim membership and hijack the whole show, for example.
Another group that is on-line, and so very different from a church group, has also been asking the question of membership. In the case of a Facebook group, anyone can be "added" by a friend, or can request to join and be added by someone who is already a member. In this case, membership is defined by a connection with someone who is already in. The title of the group also signifies something, but not everything, because inevitably some people join the group who don't meet (technically) the title of the group.
Churches are typically like this also, with many people on the margins, inquiring, seeking, moving away or towards the center.
Which is to say that both of these groups tend to operate with a "bounded set" approach to membership. You know you are in if certain practices or definitions get you across the border. You are out if they don't. Members have given. Non-members have not. And so on.
However, the concept offered by some emerging church communities, and an alternative in the world of set theory, is the concept of a "centered set." Below is a quote from the Wikipedia entry on Emerging Church. Before you read it, I'd like to highlight some concepts that are particularly important to me in my current thinking. First, I love the idea that a centered set is all those who are moving towards the center. So for example, in the Facebook ELCA Clergy group, although the group is called ELCA Clergy, we also have seminarians who participate (a group that is moving towards being clergy), as well as staff from our denominational publishing house (who are moving towards clergy with print resources). In a church, in addition to membership meaning "moving towards Jesus," you could imagine a centered set church that had a set of core practices and invited everyone to be moving towards them, without demanding a jumping of hurdles into full verses non membership. In this construal, those who are moving towards weekly worship, daily prayer, working for justice in the world, baptism, the table, etc. simply are centered and moving towards the center. Those who are moving away are not. But the outer boundaries are fluid, which allows much more grace for those on the edges (the homebound, seekers, people with anxiety about group participation, and so on).
Second, I love the idea that because centered sets are focused on the center, they simply aren't distracted by diversionary issues like who is or is not a "member." Instead, they are focused on the center itself, and there is considerable egalitarianism concerning differing opinions, approaches, etc. Even authority itself is the center, rather than some kind of hierarchy.
This is a useful way to think about almost any organization you can imagine, and let the idea help re-imagine more faithful ways to be community together in a centered, relational, fluid world.
The movement appropriates set theory as a means of understanding a basic change in the way the Christian church thinks about itself as a group. Set theory is a concept in mathematics that allows an understanding of what numbers belong to a group, or set. A bounded set would describe a group with clear "in" and "out" definitions of membership. The Christian church has largely organized itself as a bounded set, those who share the same beliefs and values are in the set and those who disagree are outside.
The centered set does not limit membership to pre-conceived boundaries. Instead a centered set is conditioned on a centered point. Membership is contingent on those who are moving toward that point. Elements moving toward a particular point are part of the set, but elements moving away from that point are not. As a centered-set Christian membership would be dependent on moving toward the central point of Jesus. A Christian is then defined by their focus and movement toward Christ rather than a limited set of shared beliefs and values.
John Wimber utilized the centered set understanding of membership in his Vineyard Churches. The centered set theory of Christian Churches came largely from missional anthropologist Paul Hiebert. The centered set understanding of membership allows for a clear vision of the focal point, the ability to move toward that point without being tied down to smaller diversions, a sense of total egalitarianism with respect for differing opinions, and an authority moved from individual members to the existing center.