Talk:Transitive set

message: A set A in not transitive iff ∈ is transitive on A. Example 1: A={0,{0},{﻿{0}}} is a transitive set, but ∈ is not transitive on A. Example 2: ∈ is transitive on A={﻿{0}}, but A is not a transitive set. -- EJ 13:18, 17 August 2005 (UTC)
replies: text: My mistake.  I was trying to find justification for using the same word -- possibly historical justification. -- Arthur Rubin 14:06, 17 August 2005 (UTC)
replies: text:  Well, the conditions look similar, at least if you strip all quantifiers and not bother much what letter denotes what. I am not sure if there is a more coherent justification of the name. -- EJ 14:24, 17 August 2005 (UTC)
thread_title:  Transitive set vs. transitive relation 

message: Explaining my revert:  A urelement cannot be a subset of anything because it is not a set. -- Arthur Rubin 19:17, 24 September 2005 (UTC)
thread_title:  Urelements and subsets 

message: Could this be a good place to elaborate on transitive closure rather than creating a separate article for it and have to repeat most of the context?  If so a redirect will be needed to this page from Transitive closure.  Vonkje 16:15, 8 September 2006 (UTC)
replies: text: You are welcome to elaborate on transitive closure here, especially because the current article is just a short stub. However, a redirect would be  problematic, as transitive closure already exists, and describes a different but presumably more common usage of the term. -- EJ 17:05, 8 September 2006 (UTC)
replies: text: I added a disambiguation link from transitive closure to this article back in April 2006. JRSpriggs 07:05, 9 September 2006 (UTC)
thread_title:  Transitive closure 

message: It would be quite interesting were transitive closure a universal object or other 
category construction. Does anyone know?Rich 01:34, 9 September 2006 (UTC)
replies: text: I doubt it is possible. Transitive sets and transitive closures are hopelessly non-invariant under isomorphism in any category I can think of. -- EJ 23:43, 9 September 2006 (UTC)
thread_title: Category Theory and Transitive Sets?

message: Could some examples of transitive sets other than ordinals be added to the article?  Thanks.
70.90.174.101 (talk) 04:00, 15 September 2009 (UTC)
thread_title:  examples? 

message: :"In the superstructure approach to non-standard analysis, the non-standard universes satisfy strong transitivity, see (Goldblatt, 1998, p.161)."
To : What does this mean? Even I have no idea what you are talking about. What is the "superstructure approach"? What is "strong transitivity"? You need to either define your terms or remove this sentence. And why is the sentence relevant to this article? JRSpriggs (talk) 02:00, 18 January 2011 (UTC)
replies: text: The universe of internal sets is strongly transitive.  I will get back to this.  Tkuvho (talk) 06:23, 18 January 2011 (UTC)
thread_title:  Unclear sentence on non-standard analysis 

message: Currently the first statement reads as follows:
In set theory, a set A is transitive, if
    whenever x ∈ A, and y ∈ x, then y ∈ A
Would it not seem more natural to the notion of 'transitive relation < on a set' to write
    whenever y ∈ x, and x ∈ A, then y ∈ A
when defining 'transitive set A'? Said another way, does it really matter in set theory which way the two parts of the logical AND are written from left to right, and if not, why not write it as I did, which seems to me more natural and illuminating, and does writing it that way violate the set theoretic idea of a transitive set? Would someone please explain?
In Naive Set Theory, p.47, Halmos, P.R. says, "Sometimes a set with the property that it includes (set inclusion symbol) everything that it contains (∈) is called a transitive set. More precisely, to say that E is transitive means that if x ∈ y AND y ∈ E, then x ∈ E."
In Elements of Set Theory, by H.B. Enderton, p.71, DEFINITION: A set A is said to be a transitive set iff every member of a member of A is itself a member of A:
                    x ∈ a ∈ A → x ∈ A.
To get to that definition, the author began with von Neumann's proposed method for representing the natural numbers with sets, and appealed to the Peano Postulates.
After stating that definition for transitive set, the author went on to state a couple of related theorems:
Theorem 4E: For a transitive set a, the union of (a union {a}) = a.
Theorem 4F: Every natural number is a transitive set.
Theorem 4G: The set ω (where ω is the set of natural numbers) is a transitive set.
Stated another way this theorem says "Every natural number is itself a set of natural numbers." and further says "We will later strengthen this to: Every natural number is the set of all smaller natural numbers."
 
In McGraw-Hill Dictionary of Scientific and Technical Terms, Fifth Edition, p.2057, "transitive relation [Math] A relation < on a set such that if a < b and b < c, then a < c." 
 
Langing (talk) 20:03, 30 June 2011 (UTC)
thread_title: Lead