The Church: invisible
The invisible church:
The word church is used to denote those "called out" of the world, as Berkhof notes in his Summery of Christian Doctrine. This can mean both those in the physical, local church or it can refer to the church in its essence -- those who throughout all ages are the spiritual body of Christ, in which there is no unbeliever, hypocrite or false confessor.
It is important to mention that we cannot say these are two different churches. That would be one way of confusing the terms. The only point of the word "invisible" is to recognize the fact that as a sin-tainted institution the visible body or local church is not perfectly able to reveal all of those written in the Lamb's book of life.
From what I can see, some of the reasons many bristle at this is because of its misuses. For instance, if someone is struggling with assurance, it is not helpful for them to ponder whether they are "one of the elect" or "part of the invisible church." These are questions that belong to the secret will of God and not helpful to our faith and practice. The facts and promises of God's REVEALED will (scripture) are enough to give us the boldness to come before the throne of grace, knowing full well that if we confess our sins, He is faithful and just to forgive us and cleans us.
Am I elect? Kind of a pointless question, just like asking if I am a part of the invisible church.
This doesn't make the term "invisible church" an erroneous term. It simply recognizes that God effectually calls out of all eternity an elect people which we as fallible men cannot know.
The Westminster Confession is one of the sources that uses this term. Here is how it reads: "The catholic or universal Church, which is invisible, consists of the whole number of the elect, that have been, are, or shall be gathered into one, under Christ the head thereof; and is the spouse, the body, the fullness of Him that filleth all in all." (CHAPTER 25, section 1)
Personally, I don't see why this should be problematic. People can misuse any doctrine. People can overlook the reality of the covenant just as easy as they can overlook the doctrine of election. Just because there is an apparent need in reformed churches to understand the covenant, doesn't give reason for us to cast aside other historically reformed doctrines. The fact is, one of the reasons the visible/invisible distinction is made is because the Roman Catholic Church denies any difference. For them, the visible is all who are saved. Period. You get baptized and come to mass, you're good to go. They literally claim the authority to decide who is saved. The invisible church is invisible simply because we do not have eyes for what is secret.