Talk:Naive set theory

Older discussion (2002–) is at Talk:Naive set theory/Archive 1

Please Verify: (08/06/08)

"In symbols, A ⊆ B means that A is a subset of B, and B ⊇ A means that B is a superset of A."

This seems wrong to me, since the direction of the symbol and the elements are transposed in the second phrase. A⊆B is probably the same as B⊇A, but not the same as B⊆A or A⊇B.

Please Check!!! Thanks ~~user:lenehey

I think it is right now, but I think it is very confusing. Could we change to something like: "In symbols A ⊆ B means that A is a subset of B and B is a superset of A, whereas A ⊇ B means that A is a superset of B and B is a subset of A." (Still me.) Lenehey (talk) 19:01, 6 August 2008 (UTC)

Another Mistake?

In the 2nd paragraph of the section entitled 'Subsets' is the sentence:

Some authors use the symbols "⊂" and "⊃" for subsets, ...

The symbols "⊂" and "⊃" appear identical to me. Is this a mistake? —Preceding unsigned comment added by 115.166.28.14 (talk) 07:07, 19 April 2009 (UTC)

If those symbols appear identical to you, there's a mistake in your browser or your fontset. How *do* they appear? If they both look like a little box, or a question mark, or some nonspecific thing like that, it probably means you just don't have a font that can render them. --Trovatore (talk) 08:17, 19 April 2009 (UTC)

Yes, they looked like boxes. I was using Internet Explorer. I have viewed the page subsequently in Firefox and can see that they render differently (like different facing sideways letter 'U'). Sorry for the distraction! —Preceding unsigned comment added by 115.166.28.14 (talk) 05:53, 23 April 2009 (UTC)

Two meanings

I realized very recently, thanks to conversation with another editor, that there are two meanings of "naive set theory". The first is the meaning intended by Halmos in the title of his book: a consistent, informal analogue of axiomatic set theory. The second meaning refers to the "naive conception of set" as any collection of objects that satisfy a well-defined property; this is the sense people mean when they say "naive set theory is inconsistent". This article is certainly about the former meaning of the word. I'd like to be a little more clear about this in the "requirements" section, but for the moment I just added a note while I think about how that section could be phrased. One option would be for me to write an article Naive concept of set and then say here that this is not what is intended. — Carl (CBM · talk) 20:02, 19 April 2009 (UTC)

It's rather worse than that, and frankly I don't like the way content is assigned to article names at all. See the "Formalist POV" section in the stuff you archived, and my subsequent proposal (which I never got around to trying to implement -- I think you did a bit of it at some point, but matters are still not satisfactory).

The name "naive", for non-formalized set theory pursued at the research level, is bad because you don't expect active research to be "naive", even if we can then quibble about how some (by no means all) workers use the word in a non-pejorative sense.

Because of that, the division of content gives the impression that workers in set theory no longer accept that there is a clear intuitive notion of "set" to which the axioms must conform, and instead have adopted the axioms themselves as primary.

I have other objections to your formulation of "any collection of objects that satisfy a well-defined property", since according to the contemporary realist approach, the objects satisfying certain properties simply cannot be "collected" at all. That being the case, extensions of these properties do not enter into the "any collection of objects" part of the phrase, and so do not cause inconsistencies. What you really mean is more something like "given any well-defined property, the collection of objects that satisfy it", where the inconsistency can be traced to the existential import of the word the. --Trovatore (talk) 21:00, 19 April 2009 (UTC)

I want to temporarily ignore all philosophical issues to just look at the overall organization.

We did merge axiomatic set theory and set theory at some point. The basic stuff is in set (mathematics), which does seem to overlap a lot with the basic stuff here.

One option would be, then, to merge the elementary set theory from this article to set (mathematics) and then reduce this article to just discussing the various meanings of "naive set theory" and point the readers to the other articles for the actual material about sets. We could use Halmos' preface to explain what he means by "naive", and I have some other citations for the other sense. — Carl (CBM · talk) 22:19, 19 April 2009 (UTC)

I noticed in the archive someone pointed at fr:Théorie naïve des ensembles. The lede to that article does head in the direction I am proposing. But the google translation really butchers it. — Carl (CBM · talk) 02:32, 20 April 2009 (UTC)

I support the idea of making this article a discussion of the different meanings of Naive set theory, and leaving actual treatment of sets to set (mathematics). The duplicate content is unnecessary. Cliff (talk) 16:20, 4 May 2011 (UTC)

change operation into abstraction?

In section 1 paragraph 3 of the article, "As it turned out, assuming that one can perform any operation on sets without restriction leads to paradoxes such as Russell's paradox and Berry's paradox.", shouldn't the "perform any operation on sets" be "define sets by unrestricted abstraction"? voidnature 08:48, 19 June 2011 (UTC)

Cantor's and Russell's paradoxes

Important note: Restriction on the scope of diagonal argument is set using two absolutely different proof techniques. Along with this restriction one of the proof techniques analyzes contradictory equivalence (R ∈ R ↔ R ∉ R) in a rather unconventional way and resolves it. Cantor’s and Russell’s paradoxes are resolved. https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B_tihhgZ1L4wMjUxZDI5NjUtYTZlMy00NDJhLWJjN2MtMDAzNDUzOWQ2Y2Ew&hl=en The paper is in English (according to a professional mathematician having position in USA - quite readable English), though, of course, it is not English of an English speaking person.

So many efforts were made by many people to work out ways around the paradoxes. The thing is done now. They are resolved. All needed now is just public support.

Thanks a lot in advance, guys! — Preceding unsigned comment added by Daniil Teplitskiy (talk • contribs) 12:37, 13 July 2011 (UTC)