well select your favorite inconsistency
I already gave it. It assumes everything that began to exist had a cause. It then assumes the universe began to exist. And it uses the concept of cause which requires time to already exist.
So, the following are assumed.
1. Everything that began to exist has a cause
2. The universe began to exist.
But there are also some basic concepts concerning causality that are important here.
3. Causality always happen within time. In particular, a cause of an event is always prior to that event in time.
4. Time itself is part of the universe.
Now, both 3 and 4 are fundamental aspects of causality and of time as we now understand it. But 3 already shows that time *cannot* have a cause. That is because any cause of time would have to be prior to time *within* time.
Hence, if time had a beginning, it is an example of something that had a beginning and is not caused.
Next, if the universe had a beginning (postulate 2), then time, which is part of the universe also had a beginning. Thus, postulate 1 would be violated.
Now, the difficulty is that KCA is based on a metaphysics that ultimately depends on Aristotelianism. And that is a model that is 2500 years old and,, truthfully, outmoded and disproved. The ancient Greek concepts of causality are acknowledged to be no loner valid.
In particular, causality is something that can happen as part of the laws of physics, but need not always be the case. Furthermore, it is something that is time-directed, so it relies on time to already exist. Hence, in particular, there *cannot* be a cause of time.
Next, we have many models that are quite plausible that have either time finite (with no cause) or time infinite into the past (with no violation of the second law of thermodynamics), so the *logical* position cannot eliminate such possibilities.