Talk:Set theory

The exposition of history is rather outdated and uncritical.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 (talk • contribs) 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 $\omega$ would be $\omega +1$, even though it is actually $\omega$ (as can be seen from the definition of rank in Von Neumann universe, since $\omega \subseteq V_{\omega }$). 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 x∈V_α+1. --Trovatore (talk) 21:50, 21 October 2012 (UTC)

Your proposal to write "the rank of x is the least α such that x∈V_α+1" would require some reqriting of that section, since currently the V_αs 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)

Multipundit's additions

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)

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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 A^c.

--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)

There's even a strict necessity to the formal undefinability of 'set', I think. If you had a formal definition (providing consistent unambiguous necessary and sufficient conditions for being a set) you could just take the extension of that definition and diagonalize out of it. Intuitively, you'd want to recognize the result as a set. (It would be arbitrary not to, and diagonalization is even constructively acceptable.) On the other hand, if you provided a definition that would seem sufficient to capture all sets at once, such as Frege's Basic Law V, it'll be inconsistent. If we were to explain why a definition of 'set' would have to be either incomplete or inconsistent, it would require adding some explanation (maybe a lot). But since it will be transparent to those here, I'm offering it as a contribution to the discussion. In short: Not only did Cantor not offer a formal definition; Cantor *could not* have done so, nor anyone else. Sunyataivarupam (talk) 02:54, 9 January 2023 (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)

References for the claim that set theory is a foundational system for mathematics

I think these phrases must be provided with some references:

Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice.

Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept that (in principle) theorems in these areas can be derived from the relevant definitions and the axioms of set theory.

I found a useful phrase for this in K.Kunen's book (page xi):

Set theory is the foundation of mathematics. All mathematical concepts are defined in terms of the primitive notions of set and membership. In axiomatic set theory we formulate a few simple axioms about these primitive notions in an attempt to capture the basic "obviously true" set-theoretic principles. From such axioms, all known athematics may be derived.

If there are no objections, I will input it into the text. Perhaps, somebody knows other useful citations, I think it would be good to add them as well. Eozhik (talk) 17:37, 8 October 2019 (UTC)

It might be illuminating to add that, in fact, mathematical analysis, topology, abstract algebra, and discrete mathematics can be derived from a very small part of set theory - the theory of P(P(P(N))), I believe. I'll try to nail down a good citation; it's a common observation. Sunyataivarupam (talk) 02:57, 9 January 2023 (UTC)

"SetTheory/OldVersion" listed at Redirects for discussion

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"SetTheory/OldVersion" listed at Redirects for discussion

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There is a {{}} can it be removed?

There seems to be a {{}} on end of the article, should it be removed? It looks like it's remains of a previous template has been added if possible can someone restore it, I can't find what it is.

Qwea1 (talk) 08:11, 29 July 2022 (UTC)  Done --Ancheta Wis   (talk | contribs) 09:19, 29 July 2022 (UTC)

It's tendentious to say that set theory is a branch of mathematical logic

Sorry to jump on the first sentence. It's tendentious to say that set theory is a branch of mathematical logic, though it is often grouped with mathematical logic for practical purposes, and because both can be thought of as "foundational" or "methodologically" central. 1) Here's a challenge: Can one find any current standard textbook on set theory that identifies set theory as a branch of logic? Jech, Kunen, and Kanamori don't, off the top of my head. 2) I doubt whether an average working set theorist, if there is such a thing, would accept the identification. 3) The most common attitude toward logic in the standard textbooks, afaik, is that one simply posits the first-order predicate calculus and moves on. There are more sophisticated interactions between the two later, of course, but precisely as interactions, I don't think they suffice to locate set theory *within* logic. 4) In philosophy, there's a standard problem, associated with Quine, that has given rise to much debate, about the status of second-order logic. The argument is that, because second-order logic incorporates a fair amount of set theory, it's best not thought of a logic at all. This would seem to indicate that logicians, and not only set theorists, would have doubts about the identification, for their own reasons. — Preceding unsigned comment added by Sunyataivarupam (talk • contribs) 03:14, 9 January 2023 (UTC)

My take on this is that set theory may not be part of "logic" simpliciter, but it's conventionally part of "mathematical logic", which is a term for a collection of mathematical topics with a historical connection to logic. The four branches of mathematical logic are typically taken to be model theory, proof theory, computability theory (originally called recursion theory), and set theory. Category theory and universal algebra could reasonably be added, but for some reason typically are not.

This grouping should not be taken to imply that the propositions of set theory are reducible to pure logic. Arguably the other three branches are not, either. Anyway this is a conventional taxonomy and I don't think it's particularly controversial among set theorists. It may just not mean as much as you think it's intended to. --Trovatore (talk) 05:41, 9 January 2023 (UTC)

Maybe the fifth branch to add these days is type theory? —David Eppstein (talk) 06:48, 9 January 2023 (UTC)

Is type theory really an independent discipline? I thought it was more a name for one of a few different formal theories. I don't think I've ever met anyone who called him/herself a type theorist.

Anyway, this seems somewhat tangential to the point at hand. --Trovatore (talk) 07:53, 9 January 2023 (UTC)

The sound-bite version is: "Mathematical logic is not the logic of mathematics but the mathematics of logic." Even that's a little dubious; there's plenty of stuff that goes on in math logic that has no obvious connection to logic in either direction. But it's at least closer. --Trovatore (talk) 17:50, 9 January 2023 (UTC)

Set theory can be interpreted to include the calculus of relations, developed by several authors but particularly by Russell in The Principles of Mathematics (1903) to dissect pure mathematics.Rgdboer (talk) 21:43, 9 January 2023 (UTC)

I guess your point is that Russell thought set theory was part of logic (as opposed to mathematical logic)?

That was not a ridiculous opinion in 1903, but it has become much more difficult to maintain in the intervening years (at least if by "logic" you mean a simple collection of clearly valid and mechanically applicable inference rules, as opposed to, say, second-order logic).

But it does not have much relevance to this discussion, because the main point is that "mathematical logic" is not a subset of "logic", but rather certain branches of mathematics conventionally grouped under that name. I suppose the Russell connection could potentially help explain how they came to be so characterized. --Trovatore (talk) 00:24, 10 January 2023 (UTC)

Advanced laws of set theory should be included

For example :-•complement laws •De Morgan's law •double complementation •empty and universal set law Yuthfghds (talk) 05:20, 27 May 2023 (UTC)

See truth table. --Ancheta Wis   (talk | contribs) 10:25, 27 May 2023 (UTC)

Frege is missing?

I am very surprised to see not a single mention in this article of Gottlob Frege. His work on the foundations of mathematics is inseparable from both the history of set theory and the development of Principia Mathematica. I will do my best to remedy this issue myself but would love others to help. Jbermingham123 (talk) 01:44, 19 September 2023 (UTC)

I don't think much of Frege has survived in terms of modern set theory, except for making an error for Russell to instructively refute, and then Principia Mathematica itself was basically a dead end. But sure, he's part of the history that should be presented. --Trovatore (talk) 04:25, 19 September 2023 (UTC)

History of set theory

In the article Indian logic it is mentioned that the school Navya-Nyāya had anticipated the development of modern set theory before Georg cantor so is it better to give credit to that school rather than Georg cantor? Myuoh kaka roi (talk) 05:12, 14 November 2023 (UTC)

Power set not a binary operation

Under Basic Concepts and Notation, the Power set of a set is listed as a Binary operation on sets. However, the power set P(A) of set A is the set containing all subsets of A. If the power set were a binary operation, it would map two sets to a third, rather than being a set which describes a single other set. GrantShirley02 (talk) 20:32, 19 November 2024 (UTC)

Good catch. Poor wording in the article. I don't immediately see a smooth way to fix it without rewriting the whole section or introducing awkward subordinate clauses. If you have a good fix in mind please go ahead. --Trovatore (talk) 20:36, 19 November 2024 (UTC)

I think my fix flows well, please let me know if it looks clunky or reads weirdly. GrantShirley02 (talk) 20:55, 19 November 2024 (UTC)