The Kalām Cosmological Argument attempts to prove the existence of God by appealing to the principle of universal cause. Kalām arguments try to demonstrate (1) that the existence of an actual infinite (a concept from modern set theory) is impossible and (2) that even if it were possible, the universe itself is not actually infinite and hence must have had a beginning.
The argument for 'first cause' originated with Plato and Aristotle, and was revived by Maimonides, Aquinas, and Ibn Russhd (Averroës). The argument is expressed in Thomistic and Leibnizian forms. It has recently been reworked by Christian apologist, William Lane Craig.
It is named for Islamic 'religious science', kalām, and is a dialectical* argument. It relies on the premise that the universe had a finite beginning (does not extend to infinity in the past), which is postulated to necessitate there being an immaterial cause for the existence of the universe.
1. The universe either had (a) a beginning or (b) no beginning.
2. If it had a beginning, the beginning was either (a) caused or (b) uncaused.
3. If it had a cause, the cause was either (a) personal or (b) not personal.
The kalām cosmological argument is thus an argument from contingency because the cosmos that exists could have been otherwise. It takes the form of a disjunctive syllogism.
P1: Everything that begins to exist (comes into being) has a cause.
P2: The universe began to exist.
Conclusion 1: Therefore, the universe must have a cause.
The second premise is usually supported by the following sub-argument:
P1. An actual infinite cannot exist.
P2. A beginningless series of events is an actual infinite
Conclusion (subargument): Therefore, the universe cannot have existed infinitely in the past, as that would be a beginningless series of events.
According to modern set theory, an actual infinite is a collection of things with an infinite number of members, such that part of an actually infinite set is equal to the whole set. This obtains because an infinite set of numbers contains an infinite number of even (and odd) numbers as well as an infinite number of all numbers. Because an actual infinite set already contains all numbers, nothing can be added to it. A beginningless series of events in time is an actual infinite.
None of which, of course, indicates that a) the universe must have a 'cause', or b) that even if the universe does have cause, that this cause is a supernatural entity, still less that this 'cause' is the deity of the Talmud/Bible/Koran. That is, looking back at the dialectical version of the argument, the universe could have had a beginning (Big Bang) and been uncaused, or could have had a beginning and have been caused by a 'not personal' agency. There is no necessity, on the basis of the kalām syllogism, to assume that a universe with a beginning signifies that a deity is probable.
*The term dialectic (Greek, διαλεκτική) refers to controversy. Dialectical exercises involve the exchange of arguments and counter-arguments that advocate propositions (theses) and counter-propositions (antitheses). A dialectical exercise could lead to
a) the refutation of one of the relevant points of view
b) qualitative transformation in the direction of the dialogue, or
c) synthesis, combination, or syncretism of the opposing assertions.
"The Existence of God and the Beginning of the Universe" by William Lane Craig. Craig's own summary of the kalam argument.