# Talk:Tree (set theory)

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Field:  Foundations, logic, and set theory

Why is this page called infinite tree, when the given definition does not require the tree to be infinite? A better name might be "Tree (set theory)" or "Rooted tree (graph theory)" -- the latter to distinguish from the existing article "Tree (graph theory)", which refers to trees without distinguished roots.

Once a final name is picked, the link should be placed on "Tree (disambiguation)" as well.--Trovatore 16:15, 11 July 2005 (UTC)

On reflection I think my vote is "Tree (set theory)", as long as the "See also" to "Tree (descriptive set theory)" is kept. "Rooted tree (graph theory)" is kind of ugly and, more to the point, is not what people usually say, except in the unusual circumstance that they have to make the distinction with unrooted trees.

An alternative possibility would be a merge with Tree (graph theory), but the uses of the two sorts of trees are so distinct that I doubt that's a good idea. Also, while I don't know much about graph theory, I don't really think graph theorists are all that interested in infinite rooted trees; it's really more of a set theory concept, which my proposed name reflects. --Trovatore 16:45, 11 July 2005 (UTC)

Oops. Set-theoretic trees don't have to be graph-theoretic trees at all. My comments about rooted and unrooted were more appropriate to descriptive-set-theoretic trees, which do have a graph-theoretic tree structure plus a distinguished root. I'll have to think about how/whether to say that on the tree (descriptive set theory) page.

Still, this is if anything a better reason than before for the move to "Tree (set theory)", since "Infinite tree (graph theory)" is even more clearly an inaccurate description. --Trovatore 06:29, 12 July 2005 (UTC)

## (Formerly) Disputed

Well, "disputed" might be too strong a word, but that's what the template wants. I'm concerned that the given definition for "branch" doesn't require it to be downward-closed, which I would think is intuitively a requirement. Of course I'm more used to descriptive-set-theoretic trees. Are there any infinitary combinitorists who can weigh in on whether a "branch" is necessarily downward-closed? --Trovatore 23:58, 12 July 2005 (UTC)

Actually, more than downward closed. I think a branch should be a maximal totally ordered chain -- that is, it should not be possible to add another element of the tree to the branch without violating total orderedness. But I'm not sure--Kunen doesn't actually define "branch" in this context, as far as I can tell. --Trovatore 00:11, 13 July 2005 (UTC)

OK, I found the def in Jech. A branch is a maximal totally ordered subset of the tree. Removed {{dubious}} tag. --Trovatore 02:32, 13 July 2005 (UTC)

## Definition of "height"

The page says

The height of a tree is the least ordinal α such that any branch has length at most α. If we have an element x of our tree and the well ordered section of the tree smaller than it is order isomorphic to α then we say x has height α.

This is equivalent to the definition in Kunen, but it takes a little thought to be sure of that. I think I'd prefer the following:

The height of an element x of a tree T is the ordinal isomorphic to the collection of all elements of T that are less than x in the tree ordering. The height of a tree is the least ordinal α such that no element of the tree has height α

I'll look at it here and reflect before making the edit in the actual article. --Trovatore 00:58, 14 July 2005 (UTC)

## The use of greek ε for set membership

Is there a reason the introduction for the article uses ε instead of ${\displaystyle \in }$? This way, the text is very hard to read and understand. Since I'm a newbie here on Wikipedia, I don't know if there is any precendent regarding the use of set membership operator in introductory text, but I would definitely vote for a change.

Avakar 13:43, 7 December 2006 (UTC)

## Complex math deserves a hug

Yes, I hug ALL types of trees.Notorious Tree Hugger Phinneas J. Hippy 01:54, 7 October 2007 (UTC)

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## Contrary to the merging with Tree_(descriptive_set_theory).

Of course, parts of the pages overlap, but the notion of a tree in general set theory bears significance also in other contexts, eg, a definition of weakly compact cardinals. Paolo Lipparini (talk) 23:22, 24 August 2017 (UTC)