Talk:Order type

Scope of definition

Is this definition correct? Mathworld has a broader definition that applies to any total order, not just to well-orders. It seems to me that is correct, and the definition here too narrow. Am I right? Francis Davey 10:45, 30 July 2006 (UTC)

Perhaps it would be a good thing to broaden this definition to include order types of total orders. But I have a question for you -- what set shall we use to represent the equivalence class of a total order under order-isomorphism? For well-orders, we have the von Neumann ordinals as archetypes. But what do we use for other total orders? JRSpriggs 02:07, 31 July 2006 (UTC)

Surely the question of representatives is additional to the notion of a type (which works without representatives). The article has been cleaned up but little is said about other order types -- for example the order types of models of N are well understood. Francis Davey (talk) 18:26, 21 December 2007 (UTC)

I recall Thomas Forster (author of 'Set Theories with a Universal Set') explaining to me that the Von Neumann ordinals only worked as canonical representatives in some set theories and that anyone who thought an ordinal was the same thing as a the representative of an ordinal didn't understand what ordinals were at all. What you are saying suggests a similar confusion, namely the difference between the existence of equivalence classes and the existence of canonical representatives of those classes. You may have the former and not the latter (depending on your set theory). You'd have to take this up with Thomas. Francis Davey (talk) 12:08, 22 December 2007 (UTC)

By the way, I do mean "classes" here as I am sure you understand. I appreciate that in some axiomatic set theories (though by no means all) the equivalence classes won't be sets.Francis Davey (talk) 12:16, 22 December 2007 (UTC)

Most set theorists work within ZFC or some other version of set theory which does not have proper classes. So defining the order type of an ordering as a proper class causes problems for them. I think it should be defined as a set (specifically a von Neumann ordinal number) wherever possible. JRSpriggs (talk) 03:13, 24 December 2007 (UTC)

There's no problem with defining proper classes in ZFC: they just aren't always sets. Proper classes are fairly commonly used in mathematics, eg in defining the category of Sets and similar. You only need sets to represent the classes if you want to manipulate the representatives with in the theory of sets. Anyway, regardless of what one might find easier to manipulate, mathematical usage of the term 'order type' is broader than merely to ordinals and so the article should reflect that. Your usage is encompassed by the usage here. Francis Davey (talk) 11:29, 24 December 2007 (UTC)

Changes to the definition

A user very helpfully broadened the definition of order type to make it clear that it does not apply to only to well-ordered sets. For some reason (that I fail to understand) JRSpriggs reverted those edits. He added the sentence:

Sometimes order type is used loosely — two ordered sets (including ordered sets which are not well-ordered) are said to have the same order type if they are order-isomorphic (also called order-equivalent).

But that is simply not true. There is nothing lose or informal about the usage of order type to describe the relationship between two ordered sets of possessing a bijection between them. This usage is common for example in the foundations of mathematics. Elsewhere in wikipedia it is given this sense. I do not, therefore, understand the edit or the reason for it. Also, I fail to see why the article should be written about ordinals, which have their own article. I will revert and ask for a discussion here. Francis Davey (talk) 11:50, 22 December 2007 (UTC)

Just so its clear I am not making this up (though I did work in the foundations of mathematics for a while and with people who really understood this), mathworld (which is a pretty good source) agrees with me. I don't have immediate access to any of the paper references as I am on holiday, hopefully another user can assist. Francis Davey (talk) 11:55, 22 December 2007 (UTC)

Thanks for reverting. Obviously, I agree. -- Dominus (talk) 22:07, 23 December 2007 (UTC)

I created this article originally to further explain the usage of "order type" in the article ordinal number. It is OK to mention that there are other ways to use it, but it seemed to me that the changes were denying (or at least obscuring) the usage for which I created the article. Notice that the article was created within Category:Ordinal numbers. JRSpriggs (talk) 03:13, 24 December 2007 (UTC)

That much I certainly didn't realise (and I am sure others may similarly not have known). I came here from non-standard arithmetic which discusses the order type of models of N - order types which are certainly not ordinals. Maybe this explains why I suggested the article be broadened way back when. I don't think the article, as it stands, should confuse anyone who wants to know about ordinals - and they can go and look at an article on ordinals if they want. Francis Davey (talk) 11:29, 24 December 2007 (UTC)

Order Type $\eta$

I corrected claim in the notation section about the usual referent of $\eta$, which is intended to be the order type of the rationals, not the integers. The mathworld site linked above by Francis Davey confirms this. —Preceding unsigned comment added by 67.194.1.187 (talk) 21:02, 24 April 2008 (UTC)

Dual order type

The definition in this article says that X and Y have the same order type if there is a monotone bijection between them. Now, the identical function is a monotonely decreasing bijection from (S, <) to (S, >), so that they have the same order type. Then every order type equals its dual and the *-notation is useless. — Preceding unsigned comment added by 81.164.106.200 (talk) 11:00, 4 February 2013 (UTC)

I think that you are confusing the issue by talking about (S, >). A better way of looking at this would be to define <_R as the reverse of <, that is, a<_Rb iff b<a. Then it is clear that the mapping from (S,<) to the reversed order (S,<_R) is not monotone increasing, but monotone decreasing which is not what we are talking about here. JRSpriggs (talk) 08:54, 5 February 2013 (UTC)

I fail to see the need to introduce a new symbol (<_R) to mean the exact same thing as >, when we already have the symbol >, which the original anonymous commenter used. I think that comment was fine as it was written, except that the article doesn't use the term 'monotone bijection', it uses 'order-preserving bijection'. I haven't checked the history; perhaps it did back in February. But the definition as it is now is fine. Hccrle (talk) 19:56, 18 November 2013 (UTC)

Rational numbers

I changed the statement about order-preserving bijections with the rationals by adding the word 'no', so it reads "with no highest and no lowest element" (the second 'no' had been missing) to clear up the meaning. Without that word, it could be interpreted to refer to an ordering with a lowest element but no highest element, which would make the statement false. I was originally just going to change the word 'and' to 'or', but this is perhaps clearer still. Hccrle (talk) 19:58, 18 November 2013 (UTC)

Thank you. JRSpriggs (talk) 20:45, 18 November 2013 (UTC)