Non-Christian worldviews cannot be internally consistent.
While the above statement is true, proving it is a different matter. However, its truth follows from the basics of the Christian faith. Since, in Christian theology, the Christian God is understood to be necessary, any deviation from that which is necessary would be impossible, entailing a logical contradiction. Specifically, to claim a deviation from that which is necessary would be to claim that what is necessary is also not necessary, at the same time and in the same way.
Pointing out why some particular detail of the Christian faith is necessary, and why any deviation from it thus yields incoherence, is one of the primary activities of the Christian apologist.
I myself have dealt with the existence of God, the absence of more than one ultimate God, the intimate involvement of God in our present moment, the omniscience of God, and the plurality of persons in the Godhead in deductive form.
Other Christian apologists such as Justin Martyr, Augustine, Anselm, Aquinas, Isaac Newton, Descartes, Pascal, Van Til, and C.S. Lewis have striven to do likewise, dealing with some of the same questions and many others as well.
Let's look at an example. Scripturally, we see that God exists in one particular necessary sense in that nothing has come into being apart from Him (John 1:3; Colossians 1:16). If things could come into being through a different mechanism, then it would not be necessarily true that nothing can come into being apart from Him. However, it is true, therefore the Biblical God is necessary for the existence of any created thing.
Thus, any view that posits the coming-into-being of anything must also posit the existence of the Christian God. Otherwise, things could come into being apart from Him, and we as Christians know this cannot happen. Other worldviews positing that things have come into being are missing a necessary prerequisite for the coming-into-being-of-things, namely, the Christian God.
It would be as though I claimed that I have a square with no right angles. You can't have a square with no right angles. That would be a contradiction.
Things can't come into being without the Christian God. That would likewise be a contradiction.
So, any view positing things-coming-into-being, and not-the-Christian-God, is a contradictory worldview, just like any worldview positing squares with no right angles.
In Christianity, however, the Christian God guarantees that all things are perfectly coherent and consistent. He cannot contradict Himself (2 Timothy 2:13; Exodus 3:14), and, as we have already seen, all of creation is His invention. This guarantees creation's non-contradictory nature as well. Because the Christian God exists, it is actually possible for humans to know things. Because the Christian God exists, humans can engage in science, math, and philosophy, all of which would be impossible apart from Him. In the Christian God are all the treasures of wisdom and knowledge (Colossians 2:3).
Circularity at its absolute finest.
The idea of something making sense to us and it being truly coherent are not quite the same thing. When I watch a movie, it doesn't occur to me that none of the characters have used a restroom in three weeks. It doesn't seem impossible at all while I'm engrossed in the movie. It's entirely reasonable, but only because I'm not really thinking about how the human body works. Things often seem reasonable to us when in fact they are not... because we just can't see the problems: we aren't omniscient. But our failure to see it doesn't mean the incoherence isn't there. Since the Christian God is necessary, if follows that any view that denies the Christian God is necessarily impossible. Obviously, though, those who do not recognize my premise as being true will not find cause to accept my conclusion. But I believe it does follow.
I understand that same key difference between Clark and Van Til in much the same way you do. Clark seems to have asserted that coherence is the thing to shoot for, and Christianity is coherent. Van Til seems to have asserted that coherence is the thing to shoot for, and ONLY Christianity is coherent.
But... take a look at this argument:
P1. Worldview X asserts that it matches reality.
P2. Worldview X asserts Y.
P3. Reality is not-Y.
C1. Worldview X therefore asserts both Y (from P2) and not-Y (from P1 and P3 combined).
C2. Worldview X is therefore incoherent.
By this logic, every false worldview is incoherent. If Christianity is true, then every worldview that contradicts it is false, and thus ultimately incoherent. Since I believe Christianity is true, I am therefore obligated to see all contradicting views as incoherent, even if I cannot myself find the flaw.