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 Definition Cardinality (size) of the set of all natural numbers. [d] [e]
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Completely rewrite:

Avoid duplication of countable set (for basic explanation)
Technical material should go into cardinal number for context
Removed a general paragraph which does not fit here:
"Greek mathematicians first grappled with logical questions about infinity (See Zeno and Archimedes) and Isaac Newton used inadequately defined 'infinitesimals' to develop the calculus; however over centuries the word infinity had become so loaded and poorly understood that Cantor himself preferred the term transfinite to refer to his family of infinities."

Peter Schmitt 22:55, 11 June 2009 (UTC)

Something missing?

There is a lot more to say about alephs, but I think that this belongs to cardinal number where it can be treated in context. Peter Schmitt 22:02, 17 June 2009 (UTC)

I agree. I did some copy-editing, mainly because I found the first sentence too complicated. I also tried to highlight the link to "countable set". Feel free to undo if you wish; I don't know much set theory. -- Jitse Niesen 09:35, 18 June 2009 (UTC)
Thanks, I did not check for links, and I wanted to avoid a "See this article" for stylistic reasons.
However, I am still thinking about the first sentence. It is nit-picking, and "we mathematicians" do not need to be reminded that aleph-0 and alike are arbitrary names and symbols, and not the mathematical object itself. But, for example, is the set of natural number, or isn't it only a symbol that can (and is) sometimes changed, and could mean another mathematical object. ω is used for the smallest ordinal number (which, in some models, is the same set as aleph0, and the same set as the set of natural numbers), but could, in another context mean an angle.
Peter Schmitt 16:44, 18 June 2009 (UTC)
Ah, so that is why you wrote "traditionally". That confused me. I see your point, but the article on Paris also starts with "Paris is the capital of France", even though one could also see "Paris" as an arbitrary name, which can even refer to other things (Greek hero, various cities in the US). I think it would be over the top to write "Paris is a string of letters which often refers to the current capital of France." But going back to this article, would you prefer it if we wrote "aleph-0 denotes …" instead of "aleph-0 is …"?
You are right that the "see this article" looks weird, so I removed it again. Let's hope that the bold font is enough to attract attention to the link.
Organizationally, I'm now wondering whether we need separate articles on cardinality and cardinal number. What should go in which article? Also, as you say, many properties of aleph-0 can best be treated in a more general article, either on cardinal numbers or more specifically on the aleph-numbers. That begs the question, do we need a separate article on Aleph-0 at all? -- Jitse Niesen 17:45, 19 June 2009 (UTC)
I am aware that the name/object problem is not special to mathematics. But it is more significant in mathematics because we define (and often change) our objects, names, etc. "Paris" will stay "Paris" (at least, unless it is officially renamed like Bombay). And this certainly is not obvious to non-mathematicians. (I am not at all sure that all books on cardinal numbers use the aleph notation.) You certainly have already observed that -- if mathematics is shown in films -- more often than not there is the formular a2 + b2 = c2 on the blackboard (but without reference to any triangle) -- as if a. b and c had fixed meaning.
But well, I may be too cautious.
Of course, one could redirect aleph-0 to cardinality (and I did not create it, I only rewrote it), but I think that it is better to have short pages which try to explain a term as simple as possible (and independently) instead of hiding it in a long (and more technical) article which probably is not read.
About "cardinality" and "cardinal number": I think that "cardinality" should be a general discussion of the notion, while "cardinal number" should be a (more technical) discussion of the concept of cardinal number, their arithmetic, etc.
Peter Schmitt 00:09, 20 June 2009 (UTC)