Talk:Finite set
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Wait a minute!!
This encyclopedia defines the terms "natural number" and "finite" in terms of each other!!
-- Anon
This isn't Bourbaki; we're not trying to develop a single well-grounded development of mathematics but instead to describe all of the various interrelatinships between mathematical (among other) concepts. So if X can be defined in terms of Y while Y can be defined in terms of X, then both definitions should be included.
That said, any interesting definitions of X or Y that don't mention the other property should also be included! If you have any good ones, then please add them.
-- Toby Bartels 06:46, 25 Sep 2003 (UTC)
I don't agree : even without being Bourbaki, if one definition is given in terms of the other and reciprocally, then it is mathematically completely useless !
Also, if n=0 is a natural number, then is {1,2,3,...,n} the empty set?
In fact, I wanted to know if according to wikipedia, a finite set can be empty (i.e., if the empty set is finite or not).
— MFH: Talk 23:29, 19 Apr 2005 (UTC)
Well, there exists no bijection between the empty set and one of its proper subsets, since there are no proper subsets of the empty set, so by default it is finite. bananaclaw May 5, 2005
First, natural number is not defined in terms of finite: neither Peano axioms nor the definition of von Neumann representation mention finiteness. Second, finiteness can be easily defined without using natural numbers. Dedekind's definition (having no proper equivalent subset) is mentioned in the article. There is also another definition which does not depend on the axiom of choice (I forgot who invented it, maybe Kuratowski?): a set X is finite iff every nonempty set of subsets of X contains a maximal element wrt inclusion.
As for n=0: empty set is finite, according to everybody, not just Wikipedia. Yes, {1,2,3,...,n} is a sloppy notation for the empty set in this case. -- EJ 18:28, 14 August 2005 (UTC)
[edit] Finite should be a dab page
Adjectives make bad article titles. I've moved what was at finite to finite set, but lots of the links are not in fact about finite sets, but about other notions. There should really be a disambiguation page at finite that would point also to other notions of finite quantities. --Trovatore 20:41, 25 October 2005 (UTC)
- Done (but it should have lots more links) --Trovatore 22:15, 26 October 2005 (UTC)
[edit] Axiom of Choice
The reason I changed the line about AC being true to assuming it is that it strikes me as very bizarre to call an independent axiom either true or false. I suppose this is a formalist bias on my part, but suggesting that it actually has some as-yet unknown truth or falseness similarly reflects a strong realism, while simply making mention of an assumption gives a more neutral perspective, I think. --Fell Collar 18:52, 12 May 2006 (UTC)
- Well, if you're going to phrase it that way you have to be consistent. If the hypothesis is to be stated formalistically (as "what we assume"), then the conclusion needs to be so stated as well, in terms of what we can prove rather than what's true. A bare mathematical statement is equivalent to the statement being true, not provable.
- It could be phrased something like "the following are provably equivalent in a formal theory including the axiom of choice". But I think that's just awkward; the usual convention is to state things Platonistically, and let formalists reinterpret as they will. --Trovatore 20:36, 12 May 2006 (UTC)
