Definition talk:Cardinal Number

Notation
I would like to adopt the notation $S$ with a double bar over it, so as to distinguish it from Definition:Cardinal. How do you make this notation? --Andrew Salmon (talk) 00:45, 31 August 2012 (UTC)


 * Don't know, try googling for it.


 * A google search has revealed nothing. $\bar{\bar S}$, which is what one person recommended, looks horrible.  I'll just stick with the $\operatorname{Card}$ notation. --Andrew Salmon (talk) 06:56, 31 August 2012 (UTC)

How will the merge be made? The definition Definition:Cardinal is a more general definition that specifies some general properties. This definition is specifically for a definition of cardinal numbers as ordinal numbers. Shall we have a specific definition under a subpage (maybe Definition:Cardinal/Cardinal as Ordinal)? --Andrew Salmon (talk) 06:12, 31 August 2012 (UTC)


 * Okay, then don't merge them. But they have to be linked somehow, either by a disambiguation page or by an also see, and an effort needs to be made to explain why these two completely different approaches are confusingly given the same name. As it stands there is considerable cause for confusion. --prime mover (talk) 06:19, 31 August 2012 (UTC)


 * I have linked to here from Definition:Cardinal. However, that page does not really define anything. It just states 'there exists a set $\operatorname{Card} \left({S}\right)$ such that $S \sim \operatorname{Card} \left({S}\right)$'; this is not a definition, at best a theorem.
 * It is definitely not the case that $\operatorname{Card} \left({S}\right)$ consists of all sets $T$ with $S \sim T$, because this is generally a class (also, check out the empty set). I think that it is a good idea to merge the two approaches as they are sort of disjoint. They amend each other. --Lord_Farin (talk) 07:13, 31 August 2012 (UTC)


 * What's wrong with $\left|{S}\right|$? Ultimately, for practical purposes, we are just defining "the number of elements in $S$". The method we choose to use to get there, or the notation one uses when one has got there, are surely immaterial. And if, during the course of an exposition, it is necessary to be precise about exactly which particular interpretation of "cardinal" is being used, then at that point one can explain exactly which one is in question. After all, once the work has been done underpinning the result that demonstrates "all these things mean the same thing, really" one does not need to distinguish between them and the precise nuances in meaning can be effectively ignored. --prime mover (talk) 08:05, 31 August 2012 (UTC)


 * Cardinality does induce an equivalence. But this is different from Cardinal, which is a particular set used to define cardinality. A precise definition is necessary to demonstrate the concept exists. I agree that when the rigorous foundations are in place, we can dispose of the actual form of the thing and use only those properties we find useful. But at the moment there are no rigorous foundations (except for this page, but it isn't merged with Cardinal yet). --Lord_Farin (talk) 09:46, 31 August 2012 (UTC)

Attempted resolution

 * 1) The page Definition:Cardinal can stay, but made into a theorem whose proof consists of a reference to Definition:Cardinal Number as the example of such a set-equivalent set.
 * 2) On Definition:Cardinal, the note to the effect that a cardinal is an equivalence class of all sets of the same cardinality can be sidelined as a "also defined as" with a warning that this is a non-rigorous approach.
 * 3) Definition:Cardinal Number can be expanded with a few words of explanation (e.g. an "intuitive" view) as the "number of elements in a set".
 * 4) The page Definition:Cardinality, which also specifies an equivalent set, but as a subset of $\N$, can be amended to match the example as defined in Definition:Cardinal Number, the same way as Definition:Cardinal will do. It should even be appropriate to link directly to Definition:Cardinal Number as a justification for its definition.
 * 5) Having done that, links to Definition:Cardinal should be replaced with links to Definition:Cardinal Number.
 * 6) Finally, the notation $\left|{S}\right|$ can be used for the notation, and that awkward double-overline construct can be mentioned in an "also known as" as a notational variant.

If this is not an acceptable approach, then I will leave this area in your capable hands to resolve, as I am otherwise out of ideas. --prime mover (talk) 10:37, 31 August 2012 (UTC)


 * Very acceptable, I'm fully supporting it. --Lord_Farin (talk) 11:01, 31 August 2012 (UTC)

On?
I don't see "On" defined anywhere. What is it? --GFauxPas (talk) 17:39, 31 August 2012 (UTC)


 * Good call. Means the set of ordinal numbers. Ongoing bad habit which needs to be broken: not linking to definitions. --prime mover (talk) 18:12, 31 August 2012 (UTC)


 * Pm tried to say: 'class of ordinals'; it's all intricate in this realm. --Lord_Farin (talk) 21:14, 31 August 2012 (UTC)

More issues
This definition formally assigns a "cardinal number" of the universal class to any non-well-orderable set. At least one theorem uses language recognizing this (something like "If $S$ is equivalent to its cardinal number, then blah blah."). There's nothing inherently wrong with doing that in a particular text, but it ends up looking very strange in a less structured context like Proofwiki. I don't know for sure, but I doubt very much that this is or ever was standard. When one speaks of cardinality without AoC, one generally says things like "If $S$ is well-orderable. . . .". Similarly, one can speak of the Scott cardinality of any well-founded set. --Dfeuer (talk) 00:55, 21 May 2013 (UTC)