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The answers here are quite mathematical. I hope somebody can explain this particular point here.

Why is it for a tree with height $\omega$ that all its levels need to be finite in order to have an infinite branch ? This question is about the necessity of the finiteness of the levels. For example what is false about the picture below ? I don't see why existance or the absence of the (encircled) subset on the right, affects the existance of the infinite branch ?

enter image description here

Definitions

A tree $(T,<)$ is a partially ordered set with $\forall y \in T:\ T_{<x} := \{y\in T|y<x\}\ \text{is well-ordered}$.

The height of an element $x$ is the ordinal $ht(x,T):=\alpha_x \cong T_{<x}$, i.e. of the order-type of $T_{<y}$.

The $\alpha$th level of the tree is $T(\alpha) = \{x\in T|\ ht(x,T) \cong \alpha\}$

A branch is a subset of $T$ that is maximal chain

The height of the tree is $ht(T) = \sup\{ht(x,T)+1|\ x \in T\}$,

Added:

I doubt about this last definition, for if we have an $\omega$-long branch, the tree would be $(\omega+1)$-heigh. Though this definition is found also in Kunen, Set Theory, An Introduction to Independence Proofs (1992), §5 Trees, p. 68. Could anyone explain what is wrong with this ?

Physor
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3 Answers3

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Take a collection of finite trees, such that your collection contains trees of every possible finite height.

Graft them all onto a single new root node.

Then the combined tree has height $\omega$ (it certainly can't have any finite height) -- but it can't contain any infinite branch. Such a branch would have to contain one of the successors of the new root node. But that successor is the root of one of the original trees, so it cannot be in an infinite branch.

Troposphere
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A tree with a single root and infinitely many children (which all are leaves) is an infinite tree without infinite branch.

Hagen von Eitzen
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  • But that long thing in the picture isn't a branch ? it is maximal chain in $(T,<)$, I mean it has already a branch, right ? – Physor Apr 27 '21 at 10:00
  • Your particular example indeed has an infinite branch, however, as the other example in this answer shows, it's not true in general. – Berci Apr 27 '21 at 10:21
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It seems to me that you are misunderstanding König's lemma. It gives a sufficient (and not necessary) condition for a tree to have an infinitely long ray/path. Namely, if a tree is infinite (has infinitely many vertices) and finitely branching (locally finite/each node has finitely many children), then it has infinite ray/path from the root (note this is much stronger than asserting that the suprema of the heights is infinite). This doesn't contradict your example.

Couchy
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    Most likely, the exercise asks to show that local finiteness is a necessary precondition to show König's lemma. Namely, that there exists infinite trees, not necessarily locally finite, which do not have infinite branches. Or in other words, that König's lemma is false without the locally finite condition. – Couchy Apr 27 '21 at 10:26
  • Another point is that, how can an infinite tree not have an infinite branch ? It is like impossible! Something should grow to $\omega$ for the tree to be infinite, follow that thing upwards then it is totally ordered and maximal!, what is wrong about it ? – Physor Apr 27 '21 at 10:31
  • There are two factors to consider: infinite branching (infinite children for a single node), and infinite branch length (infinitely long path). If a tree is infinite, it must have one of the two. – Couchy Apr 27 '21 at 10:38
  • An infinite tree is a tree with infinite vertices. It does not assert anything about the height or the width. – Couchy Apr 27 '21 at 10:41
  • @Couchy: Careful with "it has infinite height (there exists an infinite ray/branch)". I think that might be confusing in this context -- since the very point of the exercise is that "infinite height" and "there exists an infinite branch" are not a priori synonymous, but just happen to be equivalent under the additional assumption of finite branching. – Troposphere Apr 27 '21 at 10:52
  • @Physor: How exactly do you define the "height" here? I'm assuming it's the supremum of the heights of all nodes in the tree. So for the tree to have height $\omega$ all you need is that for each level $\in\mathbb N$ there's some way to reach that level. But it may be that there's infinitely many nodes at that level, and when you reach one of them, you've limited how far further you can go from that particular node. – Troposphere Apr 27 '21 at 11:05
  • @Physor: Oops, sorry, I overlooked the definition there. Fortunately it matched my assumption :-) – Troposphere Apr 27 '21 at 11:12