The Kalam Cosmological Argument is well known: if the cosmos had no beginning, it would not require a creator. Yay, for the atheist! But then, the cosmos would be infinitely old; and, so, it would be impossible for finite events (such as all those that constitute reality insofar as we can apprehend it) to complete the infinite traversal from the infinitely distant past to any moment whatever of the cosmogonic timeline. Zeno would be pleased. There could then be no present moment, for no such present moment could ever yet have happened. Nothing whatsoever could then ever happen. But, tace Zeno, there is always a present moment, events do transpire, ergo etc. The infinity of the past is refuted by the reality of any present event (or any past event, for that matter). The cosmos is therefore temporally finite, had a beginning, so stands in need of an extracosmic cause, and so forth: God, QED.
But there is also an analogous Kalam Ontological Argument. Ontological arguments proceed from a priori premises, that do not at all depend upon a posteriori observation, such as your indisputable observation of this present moment of your experience. They work whether or not there is anything out there to be observed, or anyone to observe it.
The Kalam Ontological Argument is really quite simple. It is founded upon Gödel’s Incompleteness Theorems, which we have often here discussed. The basic notion (noticed by JR Lucas) is that, the world being intelligible, it must be ordered according to some logical calculus – or, as we usually call such things, some system of natural law. But, as Lucas points out, Gödel has proved that no such calculus can be both consistent (so that it can express no contradictions) and complete (so that it is capable of demonstrating all the true propositions that it is capable of expressing consistently). The complete demonstration then of all the true propositions that a given logical calculus L is capable of expressing without contradiction depends upon the invocation of a more expansive logical calculus, M, that includes L as a subsidiary. The same goes for M: it depends for its complete demonstration upon some N. So likewise for N, etc., ad infinitum.
What we find is that the complete demonstration of all the obvious truths of *any* logical calculus hangs upon *an infinite stack of superordinate and ever more comprehensive logical calculi.*
We are here faced with an infinity of priorities not dissimilar to those we encountered with the Kalam Cosmological Argument. To demonstrate the manifold and manifest truths of any particular logical calculus – which is just to say, the truths of any particular actual world, such as our own (so as to obtain such a world in the first place: no truths about the world ↔ no world) – we cannot but recur to an infinite stack of logical calculi.
The problem though is that no finite intelligence is at all competent to plumb or fathom, or therefore to invoke, an infinite stack of truths. Nothing less than an infinite intelligence could be competent thereto. Then for any mind to make sense of anything, an infinite mind must first have made sense of everything. So, no infinite intelligence, then no lesser intelligences; but then, no intellection whatever; and so, no worlds, at all; for, a world is among other things a system of mutual intellections.
There are in fact worlds, or at least there is one such that we know of, certainly. So there are mutual intellections; so then must there be an infinite intellection. QED.
That we can even begin to explain anything whatsoever entails God.