Terminology for classes of sets

See this discussion about the confuse terminology used here in the wikipedia ("sets", "collections", "classes", etc.)... And the problem of "set of sets" vs "set of atomic elements", and the confusion nested set collection and a set of sets with some hereditary property.

Incorrect term used for the page title

The term Nested set used for the title of this article is incorrect. It should be Nested set collection instead. There is no occurrence of the standalone term "nested set" in either Knuth's TAoCP or in the referenced paper Modeling Archives by Means of OAI-ORE. Both only mention "Nested set collection" or "nested sets" (plural). In TAoCP, the term is defined as follows (see elganzua124.github.io/taocp):

A collection of nonempty sets is said to be nested if, given any pair X, Y of the sets, either X ⊆ Y or X ⊇ Y or X and Y are disjoint. (In other words, X ∩ Y is either X, Y, or ∅.)

Informally, the term is introduced in Figure 20 (a), which shows a collection of 9 nested sets, labelled A,B,C,D,E,F,G,H,J. None of these 9 sets is itself a collection of nested sets. For instance, it might appear that A is a collection of the remaining 8 nested sets B,C,D,E,F,G,H,J but it is not the case since these sets are not elements of A but its subsets. This is in contrast to the collection of the all 9 sets which has the sets A to J as its elements - thus the collection equals the set {A,B,C,D,E,F,G,H,J}. Using the word "set" for this collection would result in the term "nested set set".

As a consequence, the current introductory sentence

In a Naive set theory, a nested set is a set containing a chain of subsets, forming a hierarchical structure [..]

is wrong. The term "nested set" only has meaning in the context of "nested set collection". As a standalone term, it probably has no meaning established.

Hundblue (talk) 17:01, 24 January 2020 (UTC)[reply]

I agree with Hundblue. The term "nested set" is meaningless. Zaslav (talk) 18:24, 14 March 2020 (UTC)[reply]

Moved! Zaslav (talk) 00:19, 29 March 2023 (UTC)[reply]