Talk:Set theory

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The exposition of history is rather outdated and uncritical.

The first sentence is: "Mathematical topics typically emerge and evolve through interactions among many researchers. Set theory, however, was founded by a single paper in 1874 by Georg Cantor: "On a Characteristic Property of All Real Algebraic Numbers"."

This viewpoint has been seriously contested, among other places in books mentioned as references. Let me suggest a look at this entry in the SEP, http://plato.stanford.edu/entries/settheory-early/ — Preceding unsigned comment added by 90.169.125.151 (talk) 17:30, 2 April 2014 (UTC)

The modern understanding of infinity began in 1867-71, with Cantor's work on number theory

This phrase is currently found in the article. What is the definite article meant to imply? Tkuvho (talk) 18:01, 1 February 2012 (UTC)

Tone down rhetoric

I tried to tone down the rhetoric in "resulted in the canonical axiomatic set theory ZFC, which is thought to be free of paradoxes. The work of analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory. Axiomatic set theory has become woven into the fabric of modern mathematics". However, my edits were reverted. Tkuvho (talk) 18:18, 1 February 2012 (UTC)

The "canonical" part is not sourced and is apparently incorrect. There is a number of set theories that are widely accepted, such as NBG for instance. The bit about "woven into fabric" is sheer hyperbole. Certainly category theory needs to be mentioned as an alternative foundation. Tkuvho (talk) 18:47, 1 February 2012 (UTC)
That text is not claiming that ZFC is the only foundation of mathematics, nor is it talking about foundations in general. I see nothing wrong with saying ZFC is the canonical set theory; it is by far the set of axioms people mean when they say "set theory". Moreover, I was surprised at the edit summmary of "per talk" when no other editor had supported the change here. — Carl (CBM · talk) 16:31, 2 February 2012 (UTC)
"Per talk" meant that the editor who reverted the original changes did not respond to my explanations. Typically this means that he accepted them. If you feel the changes are inappropriate, feel free to revert. I do object to calling ZFC "canonical". This does not conform to the use of the term in mathematics. It may be "standard" and "common" but not canonical. Can you source the claim that ZFC is described as "canonical"? Tkuvho (talk) 16:35, 2 February 2012 (UTC)
We don't normally have to source individual word choices; we can weigh the overall literature and decide if the usage gives things appropriate weight. The two dominant set theory books (Jech and Kunen) are entirely about ZFC. The set theories NBG and MK are typically mentioned only to contrast them with ZFC, and are not of much interest in their own right. NF set theory is almost unknown (beyond its existence) except to those who study it. I find that enough to justify the word choice. I also noticed that Kanamori literally calls ZFC "canonical" in one of his intros in Zermelo's collected works, but I don't think it's really important whether anyone has literally called it "canonical".
I don't see much difference between saying "standard" and "canonical"; either way, the point is that if someone says they are learning axiomatic set theory the presumption is that they are learning ZFC. — Carl (CBM · talk) 16:46, 2 February 2012 (UTC)
My objection is merely that "standard" implies a convention, whereas "canonical" implies an intrinsic reason for uniqueness, of which there is none in the case of ZFC, but I don't insist. Tkuvho (talk) 17:23, 2 February 2012 (UTC)
I am not sure how your comment about NBG addresses my point. Of course it is often mentioned in contrast with ZFC, the latter being the standard theory. Nonetheless, mathematicians do work in NBG, sometimes even without mentioning ZFC. See for example the recent article by Philip Ehrlich in Bulletin of Symbolic Logic. Tkuvho (talk) 09:28, 3 February 2012 (UTC)

Foundational debate/Category Theory

I edited once, but my edit was discarded, so I'm stating here what I dislike about the statements on Category Theory/Topos Theory: I am not happy with this formulation. It seems to imply that Category Theory can interpret those "alternatives", while set theory cannot - this is in fact not the case, all those can also be modelled inside of set theory. I'm not arguing that Category Theory is not a completely different approach, but the way it's written here seems to indicate it's superior in those ways, while this is not true. — Preceding unsigned comment added by Ftonti (talkcontribs) 18:38, 21 June 2012 (UTC)

Erraneous definition of "rank"

The article states: "The rank of a pure set X is defined to be one more than the least upper bound of the ranks of all members of X." This is false. According to this definition, the rank of ${\displaystyle \omega }$ would be ${\displaystyle \omega +1}$, even though it is actually ${\displaystyle \omega }$ (as can be seen from the definition of rank in Von Neumann universe, since ${\displaystyle \omega \subseteq V_{\omega }}$). The definition can be corrected as follows: "The rank of a pure set X is defined to be the least upper bound of all successors of ranks of members of X." I actually think that the expression "one more than" is more comprehensible to the general audience than the expression "successor", which I chose to use. However, I couldn't think of a grammatically acceptable way of expressing the corrected definition in natural language while using "one more than" rather than "successor". This is why I don't immediately correct the article, but first wait for suggestions for a better wording. Marcos (talk) 10:20, 20 October 2012 (UTC)

OK, you're right. I'll see if I can think of any better suggestions. One possibility would be just to remove the sentence, and go with the definition "the rank of x is the least α such that xVα+1. --Trovatore (talk) 21:50, 21 October 2012 (UTC)
Your proposal to write "the rank of x is the least α such that xVα+1" would require some reqriting of that section, since currently the Vαs are defined using the notion of rank.
Since the error has now stayed in the article for already more than a month after I discovered it, I now remove it using my original suggestion. If someone is willing to rewrite the section in order to make it more accessible, I would certainly support this. Marcos (talk) 14:35, 23 November 2012 (UTC)
Thanks for fixing that, — Carl (CBM · talk) 16:14, 23 November 2012 (UTC)

The huge quantity of text Multipundit is insisting on adding in the Generalizations section is massively out of proportion with the importance of these topics to set theory. It cannot possibly stay. --Trovatore (talk) 04:47, 18 January 2013 (UTC)

I agree and have reverted it again. —David Eppstein (talk) 06:36, 18 January 2013 (UTC)

__________________________________________________________

Confusion in the section on Basic Concepts and Notation

In the section on Basic Concepts and Notation, subsection on Set Difference, the use of the letter U is confusing. This section uses U in the example for set difference, and for many people, U is usually used for the Universal set. An awkward reference to the universal set is used at the end of this subsection to clarify this and in my opinion does a poor job. Why not simply use another letter at the beginning of the subsection, and keep the use of U for the universal set to avoid confusion on the part of beginners like me.

So in essence what i suggest would involve:

The replacement of U for say B in the first and second sentences, leaving the rest intact.

This would mean rephrasing and thereby simplifying the introduction to the complement of a set in the next sentence. One could simply replace

"When A is a subset of U, the set difference U \ A is also called the complement of A in U. In this case, if the choice of U is clear from the context, the notation Ac is sometimes used instead of U \ A, particularly if U is a universal set as in the study of Venn diagrams."

with

"When A is a subset of U,(U is a universal set as in the study of Venn diagrams) the set difference U \ A is also called the complement of A in U." --Jwmahood (talk) 14:53, 21 August 2013 (UTC) jwmahood

In my opinion, there are two main "issues" with that section. First, it is the only elementary section on the page. Most of the page is pitched at a pretty high level, somewhere around about 3rd year university pure mathematics. But the "Basic concepts and notation" section is pitched at the most elementary level, around about year 9 or 10 high school or earlier. So it doesn't really make much sense to simplify that section even more when the rest of the page will be unreadable to anyone who has difficulty with the use of U in the complement U \ A.
A second difficulty which I see is that the paragraph about set-complements confuses the binary complement with the unary complement, which is quite common at the extremely elementary level. But anyone who can read any of the rest of the page will just skim over that anyway.
My recommendation would be to split that paragraph on set-complements into a binary complement paragraph followed by a unary complement paragraph. Then the binary complement paragraph would use a notation like A \ B, whereas the unary complement paragraph would use U \ A and Ac.

--Alan U. Kennington (talk) 15:17, 21 August 2013 (UTC)

Set is undefined

Should this article (or a more or perhaps less "advanced" article we have) explicitly point out that what a set is is mathematically undefined? What is mathematically defined is what you can do when you have a set or a set of sets at your disposal.

A set is a collection of objects

will do for most purposes, but it isn't a mathematical definition, and, as far as I know, there is none. I think Halmos mentions this explicitly in his Naîve Set Theory, I'm sure I've read it somewhere in some reliable enough source. YohanN7 (talk) 15:23, 25 March 2014 (UTC)

Well, it's certainly true that if you take a strict axiomatic viewpoint and use (say) ZFC as the axioms, then "set" is a primitive concept and is therefore not given a "definition" in the formal sense. Since I'm not a formalist, I'm not willing to identify that once and for all with the notion of "mathematical definition", though. --Trovatore (talk) 05:47, 5 May 2014 (UTC)
Aside from "mathematical definition" being (at best) ambiguous itself, would you say that
From a strict axiomatic viewpoint, e.g. using the ZFC axioms, the term "set" is a primitive concept, and is therefore not given a definition in the formal sense.
or a similar sentence would be suitable to include in Set (mathematics)? It offers the Cantor definition. Naive set theory does too, but there I think it is fine as a standalone "definition". YohanN7 (talk) 15:46, 5 May 2014 (UTC)
Honestly, I really don't think we should be emphasizing the "set theory comes from formal axioms" POV at all. Yes, my own bias comes in there, it's true. --Trovatore (talk) 18:38, 5 May 2014 (UTC)
And I do not disagree. I just want to get in some statement (somewhere) to the effect that even in systems of formal axiomatic set theory, "set" is a primitive concept and therefore not given a definition in the formal sense. The main thrust should i m o still be that sets should be thought of the way Cantor did. Many people, quite naturally, think the term "set" is formally defined somewhere, like groups, etc. The Cantor description (essentially a collection of objects) is given the status of a definition in more than one article. At least we should qualify what "definition" means and does not mean. Unfortunately, I don't think I'm qualified to come up with anything useful.
And, to clarify what I'd like to see, for instance, even if you can formally define natural numbers in therms of sets, the way to think of them as being {}, {{}}, ..., is not what you should strive for. Likewise, the sets {}, {{}}, ..., not having a formal definition is not a logical problem in the theory. One can perfectly well think of them as "collections of objects". Now I have managed to sound whimsical enough :D YohanN7 (talk) 19:24, 5 May 2014 (UTC)

Having been recently involved in dead-end debates over what a set is, I now think it is pretty much essential to point out that nobody (not even Cantor) has delivered an acceptable definition of "set" that doesn't require a mile-long hand-waving argument to justify. "Set" is not defined in mathematics period. YohanN7 (talk) 22:10, 8 May 2014 (UTC)

Hamilton

W. R. Hamilton used the phrase "theory of sets" in the Preface of his Lectures on Quaternions (1853). At that time he was wrestling with units, compounds, and collections, in particular his efforts at "triplets" that led eventually to quaternions. See his comments at page 29 and 64 where he looks back to his work of 1835 and 1848 for set theory. Since set theory has evolved in mathematics to mathematical structures and category (mathematics), Hamilton’s essays offer early evidence of the need to discriminate between compound and collection, and early suggestions of terminology in algebra. — Rgdboer (talk) 22:09, 24 August 2016 (UTC)

Finite sets

We are told that "The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s". Finite set theory is a statement of the obvious and has always been known. — Preceding unsigned comment added by 188.39.71.98 (talk) 08:15, 1 February 2018 (UTC)

If it's so obvious, why is the union-closed sets conjecture not yet proven? —David Eppstein (talk) 08:35, 1 February 2018 (UTC)
In any case, it says "the modern study", which is primarily about infinite sets. --Trovatore (talk) 08:57, 1 February 2018 (UTC)
The conjecture mentioned by Eppstein is carefully worded so as to refer to finite sets only. — Preceding unsigned comment added by 86.1.37.70 (talk) 11:50, 1 February 2018 (UTC)
I agree, that a non-trivial problem does appear in finite set theory. — Preceding unsigned comment added by 86.1.37.70 (talk) 11:54, 1 February 2018 (UTC)
See https://arxiv.org/pdf/1512.00083.pdf All these conjectures seem to refer to finite sets only. — Preceding unsigned comment added by 86.1.37.70 (talk) 12:03, 1 February 2018 (UTC)
I'm certainly not saying there's nothing interesting to say about finite sets. But the study of them is usually not called "set theory". I think people who study this would consider themselves to be doing combinatorics. --Trovatore (talk) 22:58, 1 February 2018 (UTC)

Removed false sentence: "Modern understanding of infinity began ... with Cantor's work on number theory."

Removed the false sentence: "Modern understanding of infinity began in 1867–71, with Cantor's work on number theory." Joseph Dauben's Georg Cantor (p. 30) states: "Like his dissertation, Cantor's Habilitationsschrift reflected his early interest in the theory of numbers, though his great creation of transfinite set theory was not indebted to this early work." --RJGray (talk) 18:44, 15 April 2018 (UTC)

It might be more accurate to say it began with Cantor's work on real analysis. It's not that unusual to see editors confuse real analysis with number theory, I suppose because it does after all deal with "numbers". --Trovatore (talk) 21:26, 15 April 2018 (UTC)

Thank you for your comment. I'll work on a replacement sentence or two that changes the date and tells how analysis lead him to both his 1874 and 1883 articles. I already have written something about how his work in analysis lead to the 1883 article, which introduces transfinite ordinals (see Ordinal number#History). --RJGray (talk) 02:38, 16 April 2018 (UTC)

For now, I just corrected the old sentence to state that "modern understanding" was motivated by Cantor's work in real analysis and referenced Dauben's book. --RJGray (talk) 18:23, 16 April 2018 (UTC)