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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.

Zulusbelt
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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.

Will Jagy
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  • I don't quite quite get what you're telling me. Any of those finite paths don't reach to the infinite level, right? Thus some levels are empty, thus the condition isn't fulfilled. – Zulusbelt Nov 06 '21 at 15:53
  • @Zulusbelt I think the question says that each level contains an edge from at least one path, not that each path has infinite length. You haven't given much idea of the original wording from your book/notes and the material in the pages just before the question. Still, I have it right, "no infinite paths" is part of it. – Will Jagy Nov 06 '21 at 16:08
  • Well, a node is defined as having a level of x if the path from it to the root contains x edges. And the requirement is that each level n (of |N, does that make a difference? Is that different than infinite levels?) be not empty, thus for each n there has to be a path of length n. But if n is part of |N and thus not infinite, then your answer would produce an infinite amount of finite paths with different lengths, have i got that right? But how would p_inf not be a path of infinite length? – Zulusbelt Nov 06 '21 at 16:29
  • Each path from the center node is finite. There are paths of arbitrarily large length, but finite. Draw some pictures. Take my construction, but include just $p_1, p_2, p_3, p_4$ and stop there. What are the (nonempty) levels. – Will Jagy Nov 06 '21 at 16:33
  • Well, if there are no paths of infinite length, then the level correspondign to that length would be empty. – Zulusbelt Nov 06 '21 at 16:38
  • draw some pictures. – Will Jagy Nov 06 '21 at 16:39
  • So the non empty levels would be 1,2,3,4, I get that. I get that by the definition of the paths, they are finite. But how can you define them as finite, if there are infinite of them, and thus p_inf would have infinite length? – Zulusbelt Nov 06 '21 at 16:58