Calvinism = "God saves sinners"?

Was just listening to an old podcast from Greg Koukl, president of the apologetics ministry Stand To Reason. He said a couple of things that I found interesting (and somewhat odd). He was referring to a question from a friend about whether or not he was a Calvinist, and if so whether or not he was a "five-point" Calvinist. He claims "there is no other kind," and followed that up with this whammy: that there is only one point to Calvinism, and that is "God saves sinners."

Huh? So if I accept that (very Biblical) statement I must be a Calvinist? Do Lutherans not even exist? What about others, who don't consider themselves either Calvinist or Arminian? I think his supposition is that the five points of Calvinism must all hang together and are logically interdependent. So, if there really is only one point of Calvinism, then that must mean this:

"God saves sinners" = {"Total depravity", "Unconditional election", "Limited atonement", "Irresistible grace", "Perseverance of the saints"}

Last time I checked a lot of people believed the former but disagreed with some of the points of the latter, especially limited atonement (the idea that the sacrifice of Christ was only for the elect) and irresistible grace (that God's grace which makes belief in Christ and justification possible is irresistible to the human heart if that person is truly called). This is a result of demanding that all of your theology is like a solution to a differential equation, where the initial and boundary conditions are the data of the Scriptures, and what comes out is completely determined in a completely logical and completely understood way from this information.

I believe the Scriptures are absolutely true, and I believe that there is no contradiction in God's truth, but that doesn't preclude the possibility for mystery. If we are saved, it's because God chose us, but if not, it's because we refused him. Isn't that a contradiction? It is if you demand that God be placed in a four-dimensional straitjacket. The body and blood of Christ are really, truly present in the celebration of the Lord's Supper. Does that work? Not if you believe in some kind of rigorous 16th century cosmology that claims it contradicts Jesus sitting on the literal right hand of the literal throne of God. There are some things that I believe the Scriptures teach that are just plain irreducible to human logic. That doesn't mean they are illogical, it just means that God's ways are not always reducible to ours.

STR's podcast page is here, and the relevant program is from July 10.