I'm trying to wrap my head around the concept of an infinite tree with infinite, no empty levels (so each level n of |N contains at least 1 node), and no restriction on the amount of neighbours, but no infinite paths.
My main contention is that the level of a node is defined as the length of a path from the root to it, so it would follow that if there are infinite, non empty levels, that a node in level n would necessitate a path of the length n.
Seeing as n can be infinity, the length of the path to that node would also be infinity.
Can you guys help me understand how that is possible?
I'm aware this is a variation of König's Lemma, but i don't understand how the removal of the restriction to finite neighbours would remove the infinite path that would result out of trying to reach the infinite levels.
Best Answer
fix one central node. Let it have infinitely many neighbors, with each neighbor the first step on a finite path, call it $p_n.$ Let $p_1$ have length 1, then $p_2$ has length 2, $p_3$ has length 3, and so on.