I am going to have to wave my hands I think yes. Broadly its all in the definition of "relabel" but if you shift all the numbers in omega one to the left there is still no "end point" because there is an infinite number of them.
The problem basically arises because there's a sort of anti-symmetry in the notation {0, 1, 2, 3, ....} we know where it starts on the left but it doesn't actually stop anywhere on the right and this means it behaves differently depending whether you are trying to do things to it on the right or the left (I omitted from the explanation that actually these sets are ordered from smallest to largest). The way ... is captured formally preserves that anti-symmetry and so attempts to define operations like addition which rely on doing things at one end or the other of the number start doing strange things. To be honest, I don't know if any serious attempts have been made to find a set-theoretic construction of numbers which have infinites behaving more intuitively, my guess would be that there are other consequences which are undesirable but I don't know off hand.
no subject
The problem basically arises because there's a sort of anti-symmetry in the notation {0, 1, 2, 3, ....} we know where it starts on the left but it doesn't actually stop anywhere on the right and this means it behaves differently depending whether you are trying to do things to it on the right or the left (I omitted from the explanation that actually these sets are ordered from smallest to largest). The way ... is captured formally preserves that anti-symmetry and so attempts to define operations like addition which rely on doing things at one end or the other of the number start doing strange things. To be honest, I don't know if any serious attempts have been made to find a set-theoretic construction of numbers which have infinites behaving more intuitively, my guess would be that there are other consequences which are undesirable but I don't know off hand.