If the world were a thing in itself, Kant maintained, it would be true that all past times are given (or as he put it more cautiously, "presupposed") in the present time. The existence of the world in the past times is a necessary condition for the existence of the world at this time, or else the present is the beginning of the world and there is no necessary condition for present existence at all. Thus every time is either follows an eariler time or is a first beginning. Since the whole series of time is given (or presupposed), and since it could not be given if the series were infinite, the world has a beginning in time.
On the side of the Antithesis, if the world were a thing in itself, then an empty time is impossible. For the existence of things at any given time must have a sufficient reason, and there is no sufficient reason for the existence of the world at a time preceded by an empty time. According to Kant, this is what drove Leibniz to draw the conclusion that there is no time independently of the world. However, Kant claimed in the Aesthetic that an empty time can be thought. Given this possibility, and the assumption that the world is a thing in itself, the only option open is to claim (against Leibniz) that the world has always existed; its existence in the past is infinite.
With the substitution of transcendental idealism for transcendental realism, the situation changes dramatically. Both the Thesis and the Antithesis can be declared false. The world has no beginning because the synthesis of its existence at past times can never be completed. For every remote segment of time we reach (in a modern cases, by observing light that was emitted long ago), there is a principle of reason which impels us to seek an even more remote segment of time. This is a regulative principle of reason: that the task of synthesis may never be brought to completion.
On the other hand, the indefinite task set by reason does not reveal an infintely enduring past. For every advance made in synthesis, there is always a more, and hence we cannot construct the whole infinite series. "When a member only of the series is given, starting from which the regress has to proceed to absolute totality, the regress is only of indeterminate character (in indefinitum)" (A512/B540). This Antinomy, along with the Second, is a mathematical antinomy. It has only to do with the extension (in space and time) of the world and not with the conditions under which any of its states have come about. Each prior state of the world, or remote object, is only a limitation of the present state or local object. Even in the argument for the Antithesis, the only use of the Priniciple of Sufficient Reason is negative; if there were an empty time, there would be no reason for the world's coming into being at any one time rather than another. The time's being filled does not of itself yield a sufficient reason for anything; only rules of the understanding can do so. Antinomies which concern use of the rules of the understanding in determining the existence of objects and their states are called dynamical antinomies.
The Third Antinomy is dynamical, as it is about the causal conditions required for any change in the world. According to the Second Analogy, every change in appearances is subject to a rule which makes that change necessary given the existence of the prior state of the world. Thus, it rules out, among objects of experience, any change which occurs spontaneously, or not in accordance with a rule. The ability to bring about a change spontaneoulsly Kant entitles transcendental freedom. Thus among the objects whose activities in the world of appearances which bring about changes, none are transcendentally free. (In Kant's jargon, the "causality of the casue" is the activity of the agent bringing about a change.) No spontaneous acts can be found in appearances. This is the Antithesis of the Third Antinomy, and Kant accepted it as being true. Its argument recapitualtes that of the Second Analogy, that any violation of law would destroy the coherence of experience, leaving us unable to distinguish the objective from the subjective. The argument does not depend on regarding the world as a thing in itself.
On the other hand, the argument of the Thesis does make this assumption. The argument depends on essentially the same assumption on which that of the Thesis of the First Antinomy depends, that all the conditions are presupposed when the conditioned is given. Thus when something happens, it requires a reason for its happening, the reason being an activity bringing it about. But this causality of the cause itself happens, and thus demands its reason. In order that all the conditions be given, there must be a first member of the causal series which initiates change through its own activity, but whose causality is not dependent on any other activity (cf. Aristotle's prime mover). As with the First Antinomy, the Thesis is false, since no such thing can be met with in experience, as is shown in the argument for the Anithesis
Because the Third Antinomy is dynamical, the outcome can be different from that of the First. Specifically, in the First Antinomy it must be denied that the world has any definite span of existence in the past. Any first beginning must itself be a member of the series, and such a member cannot be found. But since the Third Antinomy is dynamical, reason can look outside the series of causal conditions in order to find a spontaneous condition for change. So a change occurring in experience as a result of a necessary cause which is also an object of experience can also be the effect of a causality standing outside of experience.
It is possible, Kant contends, that such a causality resides in the transcendental object which underlies experience. For example, through some activity of my empirical self, an act of will, I bring about a change either in my mental state or my physical body. I will to eat an apple, and to this end I raise an apple to my mouth and bite a piece from it. All this takes place in experience, and it is subject to necessary laws. In addition, my willing to eat the apple is itself an effect of some cause to be found in experience. But that act of will is also only the appearance of some transcendental object which is otherwise unknown. (That is, all that is known is that there is an object = X which is the substratum of the appearance.) It is possible that in this transcendental object there is a power of spontaneous action, and that the consequence of the action is what appears as the eating of the apple. In this way, a single event in the world of exprience might be determined in two different ways, by some prior state in time in conjunction with causal laws, and by the activity of the underlying object. In this way, an event might be both determined and freely brought about.