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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.
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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.
792:). 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.
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553:"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"?
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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".
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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
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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
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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
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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
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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
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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,
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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
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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
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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
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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
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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
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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
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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.
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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.
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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
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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
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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
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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.
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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.
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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.
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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.
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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"."
1231:, 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.
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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.
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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.
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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.
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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".
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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
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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
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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. --
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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.
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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.
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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.
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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.
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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
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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?
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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.
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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.
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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.
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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.
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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
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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.
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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. --
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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. --
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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.
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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. --
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reflected his early interest in the theory of numbers, though his great creation of transfinite set theory was not indebted to this early work." --
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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
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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,
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This would mean rephrasing and thereby simplifying the introduction to the complement of a set in the next sentence. One could simply replace
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The "canonical" part is not sourced and is apparently incorrect. There is a number of set theories that are widely accepted, such as
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The sound-bite version is: "Mathematical logic is not the logic of mathematics but the mathematics of logic." Even that's a
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I guess your point is that Russell thought set theory was part of logic (as opposed to mathematical logic)?
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For example :-•complement laws •De Morgan's law •double complementation •empty and universal set law
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until a consensus is reached, and readers of this page are welcome to contribute to the discussion.
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definition, and, as far as I know, there is none. I think Halmos mentions this explicitly in his
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mathematically defined is what you can do when you have a set or a set of sets at your disposal.
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This phrase is currently found in the article. What is the definite article meant to imply?
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And I do not disagree. I just want to get in some statement (somewhere) to the effect that
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I tried to tone down the rhetoric in "resulted in the canonical axiomatic set theory
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summmary of "per talk" when no other editor had supported the change here. — Carl
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References for the claim that set theory is a foundational system for mathematics
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of them as being {}, {{}}, ..., is not what you should strive for. Likewise, the
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It's tendentious to say that set theory is a branch of mathematical logic
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I agree, that a non-trivial problem does appear in finite set theory.
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does too, but there I think it is fine as a standalone "definition".
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Anyway, this seems somewhat tangential to the point at hand. --
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An editor has identified a potential problem with the redirect
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I found a useful phrase for this in K.Kunen's book (page xi):
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I am not sure how your comment about NBG addresses my point.
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__________________________________________________________
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Maybe the fifth branch to add these days is type theory? —
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All these conjectures seem to refer to finite sets only.
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or a similar sentence would be suitable to include in
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1911:truth table
1719:| contribs)
1713:Ancheta Wis
1301:Finite sets
1276:quaternions
347:Mathematics
338:mathematics
294:Mathematics
1997:Categories
1417:86.1.37.70
1392:86.1.37.70
1371:86.1.37.70
1147:definition
405:Archives:
202:Philosophy
191:philosophy
141:Philosophy
1957:Trovatore
1896:Yuthfghds
1876:Trovatore
1824:Trovatore
1801:Trovatore
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619:Of course
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1748:contribs
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972:Jwmahood
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1200:YohanN7
1103:YohanN7
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229:on the
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1918:(talk
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590:CBM
540:CBM
515:NBG
488:ZFC
366:Top
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