We argue that it is impossible for a beginningless body to exist, because if it did exist, then this body would either be:
- Attributed with a beginningless motion.
- Attributed with a beginningless rest.
- Attributed with a beginningless sequence of the above.
And all three cases are impossible, so it is impossible for a beginningless body to exist. Modern relativity can be used to argue for the impossibility of the second case.
Attributed With a Beginningless Motion
It is impossible for a beginningless motion to exist. A beginningless existence is one that is not preceded by anything, whereas being preceded by another is an essential feature of motion. This is because motion is the body occupying a space while having been in another space during the previous moment in time. By virtue of this prior state of occupying a different space, no single motion can be said to be beginningless.
Attributed With a Beginningless Rest
We can argue for the impossibility of a beginningless rest using relativity. So we say:
- It is impossible for a body to be attributed with a beginningless motion.
- If it is impossible for a body to be attributed with a beginningless motion, then it is impossible for a body to be attributed with a beginningless rest.
- Therefore, it is impossible for a body to be attributed with a beginningless rest.
The first premise was already proven.
As for the second premise, it is true because the distinction between rest and motion is abstract under relativity, one that depends solely on the observer’s frame of reference. Therefore, there is no extra-mental difference between the two states. So since it impossible for the state of motion to be beginningless, then it is impossible for the state of rest to be beginningless as well.
Let us use an example to show that the distinction between rest and motion is an abstract one: suppose Zayd and ‘Amr were standing still on a moving train carriage, and Layth was standing on the platform outside. Relative to Zayd, ‘Amr is at rest. However, relative to Layth, ‘Amr would be in motion.
Indeed, the fact that the distinction is abstract is even clearer when we remember that the Earth, with everything on it, is in constant motion around the sun. Even though relative to us, some of the objects on Earth are at rest.
Thus, a beginningless rest is impossible.
Attributed with a sequence of the above
Meaning: attributed with an emergent motion, preceded by another, preceded by another, ad infinitum. Or attributed with an emergent rest, preceded by another, preceded by another, ad infinitum. Or constantly alternating between motion and rest, since eternity past.
All of the above is impossible, because Tasasul is impossible.
 This argument is what we call Burhan Huduth Al-Ajsam. Another formulation for the same argument can be found here.
 Under relativity, “rest” and “motion” would be extra-mentally identical. Both signify a single accident that subsists within the body, by virtue of which the body is in the position it is in relative to others. And so, relative to some observers, the body would be in motion, and relative to others the body would be at rest.
On the other hand, beginninglessness or emergence are ascribed to this extra-mental accident that subsists within the body. Not to a relation between the accident and any observer.
Thus, if beginninglessness for this accident is deemed rationally impossible from one observer’s frame of reference, then it must be rationally impossible from all. To say otherwise, is to claim that the accident is both beginningless and not beginningless, and that is contradictory.
 Another way to demonstrate this, is to express “rest” in terms of motion. So we say: a body is at rest with respect to another, if both bodies are moving at the same speed, and in the same direction. And since motion is necessarily emergent, rest is so as well. For rest would be meaningless without motion, and so no rest can exist before motion emerges into existence.
 Since the position of ‘Amr relative to Zayd is not changing, whereas ‘Amr’s position relative to Layth is changing.