Talk:Set Theory/Sets

Would it be appropriate to begin this chapter with the following concise definition of a set?

A set is an unordered collection of distinct elements.

--Rickpock (talk) 05:41, 17 February 2008 (UTC)


 * Yes, although be sure to point out that set elements are not necessary distinct per se, it's just that repetitions are ignored so that the set {1, 1} is equal to the set {1}. Skyc (talk) 05:59, 17 February 2008 (UTC)

I made the change and continued into elements. I feel like I was stumbling over my words a bit with the elements, but I still feel it was an improvement. This is my first wiki-anything edit, so I'm likely being overly timid.

--Rickpock (talk) 20:43, 17 February 2008 (UTC)

Hausdorff's "Mengenlehre" (I only know it in the Dover reprint of the 3rd German edition and I have no idea if it has ever been translated into English) defines a set (in its second sentence) "Eine Menge ist eine Vielheit, als Einheit gedacht." This is a bit difficult to put into English in a satisfactory way. Perhaps "A set is a Multiplicity thought of as a Unity." I would like that better if it were punchier.

I think the definition of a set as "an unordered collection of distinct elements" begs the question because, in fact, it simply defines a set as a collection leaving "collection" up in the air.

There is a question of atmosphere here. If there is an branch of mathematics that should tend to extreme rigor it is set theory. Hence if we use an intuitive description of a what a set is we need to be up front about the fact that rigor has been lost.

The members of a set are conceived of, ontologically speaking, as having existence. Hence they MUST be distinct. Another way of saying that is that they are instances rather than values. It might be well to either start with a discussion of these semi-philosophical considerations or, breezing through them in the main text, to collect them into an appendix.

In most branches of mathematics I could start with postulates/axioms and go on from there. But there are no definitive axioms for set theory and, in light of Goedel's work, can probably never be. The discussion of which axioms to use is an integral part of set theory.

I would recommend Hausdorff to you as a guide except that the scope of his book is different than what people think of as set theory today. DKleinecke (talk) 16:10, 13 June 2008 (UTC)