# Talk:Countable set/Draft

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 Definition:  A set with as many elements as there are natural numbers, or less. [d] [e]
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## Title

Shouldn't this live at enumerability? And an article about countable sets should live at countability, I should think, as well. I don't know, I'm not giving an order, I'm just saying how I would do it. There's an issue to think through here. --Larry Sanger 12:45, 22 February 2007 (CST)

Actually, I think I agree with you. I should have titled this article Countability. Can you remind me how to rename an article? Thanks. --Nick Johnson 13:55, 22 February 2007 (CST)
I'm not so sure. The word "countability" is quite rare, definitely rarer than "countable". Besides, we also have countable nouns, so then the choice is between countable set and countability (mathematics). I think the current title is more natural. -- Jitse Niesen 18:22, 15 August 2008 (CDT)
• Countability" and similar page titles are against common usage in mathematics. A redirect from "Countability (mathematics)" is, of course, possible. (We say "natural number", "rational number" not "naturality" or "rationality", and it should not be "Continuity", but "Continuous function", etc. Peter Schmitt 09:32, 8 June 2009 (UTC)

## suggestions

1. I agree that a better article title would be countable (or countability).
2. The current version has the sentence: "Inductive proofs rely upon enumeration of induction variables." Not really: induction is a procedure that applies to the natural numbers and only the natural numbers. It might be that a function proving countability of a set might translate one problem into another problem for which induction is relevant, but that's not the same thing.
3. Narrative! The amazing fact about infinite sets is that there are lots of different sizes of them. An article on countability should introduce the reader to this paradoxical point of view, take them through the idea of using bijections to define "same size" (cardinality), and then discuss the role of countable sets in this hierarchy. Cantor's proof of the uncountability of the reals should of course be mentioned. Remember that the vast majority of readers will not know what a one-to-one function is nor the significance of the word "onto", as opposed to simply "to". If one needs to know the topic already to understand our article, then the article isn't going the right direction.

- Greg Martin 09:41, 20 May 2007 (CDT)

1) See above. 2) I guess it's a matter of definition; what does "induction" mean? Incidentally, there is also "transfinite induction" which works on uncountable sets. 3) Agreed, though we shouldn't reproduce the article on Cardinality here. -- Jitse Niesen 18:22, 15 August 2008 (CDT)

## Use of term "enumerable"

The article currently states that "an enumerable set has the same cardinality as the set of natural numbers." That's true about infinite sets, but isn't the word "enumerable" sometimes used about finite sets as well?

Ragnar Schroder 00:33, 29 June 2007 (CDT)

I agree. I made some changes, but it looks like more are needed. By the way, in my experience "countable" is more common than "enumerable". -- Jitse Niesen 18:22, 15 August 2008 (CDT)
"enumerable" is not equivalent to "countable"! "Enumerable" is a term from computability theory. Every enumerable set is countable, but most countable sets are not enumberable. Peter Schmitt 09:36, 8 June 2009 (UTC)

## Subpage "Examples"?

I think that this article now is more or less complete. Furthermore, I think that the "Examples" should better moved to a subpage, but I am not sure if the system "allows" this since this is not a predefined subpage type. Peter Schmitt 22:56, 20 June 2009 (UTC)

## Bibliography

I don't think that it makes sense to list here books which belong on the list for set theory. A link to that Bibliography should be enough. Opinions? (This is question which also is relevant in other similar situations.) Peter Schmitt 23:05, 20 June 2009 (UTC)