Kant presented four antinomies, each presented as a pair of thesis & antithesis.
The first says that the world does have a beginning in time, he argues by contradiction:
If we assume that the world has no beginning in time, then upto every given moment an eternity has elapsed, and there has passed away in the world an infinite series of successive states of things. Now the infinity of a series consists in the fact that it can never be completed through successive synthesis. It thus follows that it is impossible for an infinite world-series to have passed away, and that a beginning of the world is therefore a necessary condition of the world' s existence.
Priest, in his book Beyond the Limits of Thought demolishes his argument by appealing to set theory and stating that Kant's appeal to Aristotle's assertion that completed infinities do not exist doesn't hold.
As far as I see, Priest's argument doesn't hold:
Set theory does have infinities, but I'd argue this simply marks out a new iteration towards the infinite which doesn't have a completion. More importantly, set theory is conceptual, and time is a physical notion. One needs to think about the infinite past in a physical way - and this is exactly what Kant is doing by saying "[T]here has passed away in the world [...]". I can't see how it is possible to assert that time past can be infinite when we consider that time must pass. We could have, indeed an infinite number of worlds before us, with their own notion of time - but in this world with its own notion of time - time past must be finite.
Of course, with the Standard Model it is taken that time does have a beginning.
Is there a better argument as to why it must fail? Or is Priest simply presenting the standard arguments? Or is the failure of this thesis actually beside the point - seeing that Kant is working at the limits of what we can know?