Kant expresses the cosmological ideas as four distinct antinomies, or pairs of seemingly contradictory metaphysical propositions. They are:
(1) The claim that the world has a definite beginning and end vs. the claim that the world is infinite
(2) The claim that all things are made up of simple, indestructible, indivisible parts vs. the claim that everything is composite and infinitely divisible
(3) The claim that we can act in accordance with our own free will vs. the claim that everything we do is determined by nature
(4) The claim that there are necessary causes vs. the claim that nothing is necessary and everything is contingent
None of these claims can be verified in experience, and so we are tempted to think that they deal not with appearances but with things in themselves. Reason by itself seems capable of proving either side of each antinomy. Rather than come down on one side or the other, Kant proceeds to show how each antinomy results from a misunderstanding of the matter being discussed.
The mistake in the case of (1) comes from treating space and time as things in themselves rather than as intuitions of our faculty of sensibility. Space and time are features of our experience, and do not exist independently of experience. It makes no sense to ask whether or not the world has a limit in space and time, since that limit would exist outside the realm of our experience.
In (2), when we talk about the parts into which a composite thing can be divided, we are assuming that these already exist, waiting inside the composite thing. But these parts are only appearances, and so cannot have any existence until they are experienced.
In (3), causal necessity and freedom are made to seem contradictory when in fact they are compatible. The laws of nature can only operate within the confines of space and time, and so are applicable only to appearances. Freedom, on the other hand, is the ability to outside the confines of causality, and so to exist outside the confines of experience. Freedom, then, is applicable only to things in themselves.
Our faculty of reason does not deal with experience, and so we are free in our capacity as rational beings. This freedom must express itself only in general maxims that do not depend on causal influence or particular times and places. In obeying these general maxims, we still follow regular laws in the world of appearances. Thus, we can be free and also be subject to the laws of nature.
The seeming contradiction in (4) is similarly resolved if we see that one half of the proposition talks about things in themselves and the other half of the proposition talks about appearances. In the world of appearances, every causal connection may be contingent, which is to say it could have happened otherwise. Nonetheless, these appearances might have a necessary connection to things in themselves.
Kant deals very briefly with the idea of a God. In the Critique of Pure Reason, he shows at length the flaws in all the supposed proofs for the existence of God. Here, he simply points out that any "proof" of God's existence is a purely intellectual exercise, and cannot lead us to fundamental and substantial conclusions regarding the nature of experience.
In conclusion, Kant remarks that while there are justifiably many mysteries regarding what we find in experience, there should be no insoluble problems in the realm of pure reason. These problems deal only with reason itself and do not reach beyond our own minds into experience.
The four antinomies Kant presents as "cosmological ideas" are common topics of metaphysical debate. In each case, Kant applies his distinction between appearances and things in themselves in order to resolve the antinomy. In the first two, he shows that both sides of the antinomy mistake appearances for things in themselves, and concludes they are both false. In the second two, he shows that two seemingly contradictory points of view are actually both acceptable so long as we recognize that one is applied to appearances and one is applied to things in themselves.
The first antinomy assumes that space and time exist independently of our experience and asks whether or not they have any limits. The second antinomy assumes that the objects of our experience have an independent existence and asks whether or not they have fundamental, simple parts. In both cases, we are trying to extend our knowledge of phenomena we've experienced beyond our experience of them. Kant reminds us that the objects of experience are mere appearances, and that the space and time in which we perceive them are constructs of our pure intuition. In other words, they do not exist beyond our experience of them.
Both these antinomies might seem a little odd in the light of modern physics. We have found a limit to space and time in the Big Bang, and we have identified the simple parts of objects in atoms and the elementary particles that constitute these atoms. Still, Kant could point out that these discoveries have been made in the realm of physics, not metaphysics. What we have discovered are the limits of observable experience, not the limits of things in themselves. The things in themselves that are the source of these appearances exist outside the realm of space-time and scientific observation.
The third antinomy is probably the most interesting, as Kant's answer to it is his ethical theory in a nutshell. The problem of free will is an old one, and a favorite topic of philosophical debate. If we had no free will, we couldn't be held responsible for what we do: we would be able to excuse our wrongdoings by saying "I had no choice." Freedom, then, consists in having a choice, in one's actions not being predetermined by outside forces. However, nature's laws dictate that every event is caused by some previous event, and that every event in turn acts as a cause for some subsequent event. How can we be said to have free will or to act independently of outside forces, without violating these laws?
Kant's answer is that cause and effect are products of the faculty of understanding and can be applied only to appearances, while freedom is a product of the faculty of reason and has nothing to do with appearances. Because freedom has nothing to do with appearances, it is outside the boundaries of time and space. As a result, a free act cannot be contingent on the particularities of what is happening at a particular time or in a particular place. Free acts must abide by general maxims. This theory is more fully explained in Kant's Groundwork for the Metaphysics of Morals, in which he claims that free acts take the form of a "categorical imperative" which insists that our actions follow maxims that we could will as general laws. Freedom does not mean spontaneity; it means obeying our own law. Because our freedom manifests itself in an orderly, law-like manner, it does not violate the laws of nature that apply to all appearances.
The fourth antinomy deals with necessity and contingency. The question is whether things necessarily happen the way they happen or whether they might have happened otherwise. To reconcile this antinomy, Kant identifies two different kinds of causation: a contingent one that determines how causes work in the world of appearances, and a necessary one that determines how things in themselves cause the appearances we experience.
Kant talks here about things in themselves as acting as causes and as being necessary, but both necessity and cause are pure concepts of the understanding, and are thus applicable only to appearances. Kant could excuse himself by saying he does not use terms like "cause" and "necessary" literally, but simply for lack of a better expression. Language can only describe the world of appearances, and when dealing with things in themselves it is inadequate.
Kant's discussion of the idea of God is very brief, mostly because his reasoning doesn't change: ideas of reason can only help us sort things out in our heads, but they can't tell us anything substantial about the world outside our heads.