wtf

Not at all. I never studied differential geometry and your posts have gotten me reading up and trying to understand the point about the pushforwards. Apparently this is an abstract way of looking at the differential as a map from one manifold to another. The Wiki article on fiber bundles is very good. It's a hairbrush! The base space is the handle, the fibers are the bristles, there's a natural map from each bristle to its basepoint. A continuous right inverse of this map is a section, aka a vector field. This relates to ajb's recent comment on another thread. I'm finding this all very interesting. Your very lack of clarity is motivating me to try to understand. Don't stop on my account.

If the subject of the thread was, "How many properties does a flapdoodle have," then the natural response would be: "Well, first tell me what you mean by a flapdoodle." I see two cases here. * If you are interested in how mathematical sets work, I'm happy to give my perspective. * But if you only want to say that well, for all we know a set is a category or a group (when in fact is it NOT these things because these things have very specific, widely agreed on technical meanings) then I can't argue with you because you are not doing math. You're doing something else. You listed words like class, group, and category, which are each technical terms that do NOT mean set. So if you want to say, "Well if we deny the standard terminology of math, then I can make the following point," then I can't argue with you or disagree with you or frankly even conversate with you. Because I'm not interested in the point of view that says, "Let's pretend to talk math while denying and ignoring the actual content of math." I tossed out that [math]\mathbb{R}/\mathbb{Q}[/math] example for its inherent interest in showing readers how modern mathematicians use the axioms of set theory. And for people who have maybe heard of Banach-Tarski or nonmeasurable sets, a little step in the direction of those constructions. And of course IMO it does support my point that there are sets not characterized by properties. But I can already prove that via a cardinality argument so I've presented now a formal proof plus a concrete example. I really just don't get where you're coming from. If you want to say well nobody really knows what a set is so for all we know each element has exactly 47 properties, it's ok by me. That's doesn't invalidate my point Maybe I'm just not understanding you. Can you clarify your intent? I feel like I'm supposed to defend something but I actually made my point several posts ago and don't understand yours. Or perhaps ... would you say that perhaps you're struggling to understand more about the philosophy of set theory? You know quite a lot of thought has gone into all this by people far smarter than me. From Cantor through Zermelo and Turing and Gödel and Frege and Wittgenstein (who thought it was ALL wordplay) right up to the modern geniuses like Hamkins and Woodin, exploring and defining the very forefront of what humans think about sets. So are you perhaps just trying to make some point about the philosophy of sets? I can't parse that. Can you explain please? I went back to #8, you asked if the empty set has properties. Yes the empty set does have properties, lots of them. Let's list a few. * It's the set of flying purple unicorns. * It's the set of married bachelors. * It's the set of real numbers not equal to themselves. * etc. So the empty set is not an example of a set characterized by no properties. It's an example of a set characterized by MANY properties.

Xerxes, your subject matter is very interesting to me since it's just beyond what I currently know, hence I would be motivated to learn more. However, your exposition is unclear. Just to take one example, you said that the mapping from a vector space to its dual is called a "functor." Now if one knows what a functor is (from category theory) then your usage is totally unmotivated and far too specific. A mapping is a functor because it has certain properties, which you didn't mention. On the other hand, if one doesn't know what a functor is, they would be baffled. Likewise your discussion of pushforwards. You said it's something in operator theory, as if you either don't know or don't care to explain that a pushforward is a very general construction that abstracts particular ideas in many fields of math. Unfortunately the relevant Wiki pages aren't very good, and I ended up reading the section on tangent bundles in an online pdf of Spivak's big differential geometry book. At that point I understood what you were trying to get at. Remember, exposition is a separate problem from math. It's clear that you know some differential geometry but perhaps haven't given much thought to exposition. As a start, you might consider asking yourself who is your audience and what is their background. If someone has had a year of calculus and a semester of linear algebra, you could build up everything you've said from first principles, using a curve in the plane and the tangent spaces (ie tangent lines) at each point. This would be a beautiful writeup that I'd love to read. If that's your aim, it's something you could challenge yourself to do. But as it is, all you've done is toss out some random factoids devoid of context that are either badly stated (to a knowledgable reader) or baffling and pointless to everyone else. I hope you will take this as constructive criticism. I would LOVE to read a clear explication of your topics. As it stands, you have major expositional problems and people are trying to tell you that.

Hey thanks, no prob! Interesting point. NST is actually an axiomatic treatment of set theory but not excessively formal. But I definitely agree that a lot of people get quite far in math without paying much attention to the fine points of set theory. Honestly I wasn't trying to best anyone, just toss out an example of how sets can be created without lists of elements and without predicates. You responded to a question on category theory the other day so I'm sure you can best me there Anyway I'll still go with "zero" as the number of properties necessarily possessed by all the elements of a set that characterize the set, if that was the original question.

I'm surprised at your bafflement but I'll try to be more clear. In truth I have nothing to add to what I've already said, but I'll try to say it better. Not in the least. My challenge is to make myself clear. This is referring to your erroneous construction of a set as [math]\{x : P(x)\}[/math] where [math]P[/math] is some unary predicate. This is not valid set-builder notation. What it is, is informal set-builder notation. It's appropriate in high school, in non-rigorous math classes, in discrete math class, etc. And even in higher math, it's perfectly acceptable to use it, as long as we know in the back of our mind that we really mean [math]\{x \in Y : P(x)\}[/math] for some existing set [math]Y[/math]. In this thread, since the formal properties of sets are under discussion, it's important to be careful to understand exactly how sets may be formed according to the rules of set theory. The use of "set theory without being formal about it" is called naive set theory. It's a perfectly valid thing in every context where the difference doesn't matter. But if we're talking about the properties of sets, we have to work in the realm of formal set theory. We are constrained by the axioms and their logical consequences. It was believed prior to 1901 that a set could be defined entirely by a predicate. Russell's paradox falsified that belief. If you would please read the Wiki page for Russell's paradox this point would be abundantly clear to you. So, what is the EXACT rule by which we can form a set based on a predicate? This is given by the Axiom schema of separation. https://en.wikipedia.org/wiki/Axiom_schema_of_specification This schema is actually an infinite collection of axioms, one for each predicate. It says that if you have an EXISTING set, you can cut it down by a predicate. You can not try to cut down the universe by a predicate, that's Russell's paradox. I do hope you'll take a moment to glance at the Wiki links I'm giving. All I'm doing is walking through the basics of ZFC. By the way I should mention that's Zermelo-Fraenkel set theory. https://en.wikipedia.org/wiki/ZermeloFraenkel_set_theory. The C stands for the Axiom of Choice, so ZFC is Zermelo-Fraenkel with Choice. This is the standard axiom system for modern math; although the professional set theorists study exotic variants. [math]x \in x[/math] makes perfect sense. There are in fact people who study non well-founded set theories. These are perfectly consistent theories in which sets can be elements of themselves. https://en.wikipedia.org/wiki/Non-well-founded_set_theory In fact, why can't a set be an element of itself? The ONLY reason is that in ZFC, we make an explicit axiom that says that a set can't be an element of itself. Or more precisely, that there are no infinite descending membership chains. https://en.wikipedia.org/wiki/Axiom_of_regularity NOTE WELL that without this axiom, sets that contain themselves are perfectly valid, and are in fact object of mathematical study. We choose to accept the Axiom of Regularity simply because we only care about well-founded sets; not because [math]x \in x[/math] is logically impossible. On the contrary, from a purely logical point of view, one can choose one's sets to be well-founded or not. In fact it's been noted that the Axiom of Regularity is never used in proofs. It's only purpose is to ensure that we're always dealing with well-founded sets. And if we didn't make this rule explicit, there would be no reason why sets couldn't contain themselves. I hope that looking at the Wiki pages I linked will make the point clear. In fact you only need to give the Wiki page for Russell's paradox a good read. It's unfortunate that the forum software doesn't like that particular link, but please do Google Russell's paradox and give a careful read. I'm grateful for small points of agreement. Perhaps this is a good sign. I should add, though, that in fact the definition of the empty set is a bit of a controversial point, since we can't define it unless we already have some other set, and where'd that one come from? Let's leave this aside for now, but do be aware that your way of defining the empty set is not without problems. Very peculiar way of putting it. But you are confusing well-foundedness with unrestricted comprehension. This is a very common false belief, that Russell's paradox says a set can't be a member of itself [or your alternative phrasing, if that's the same]. Rather, that is the principle of well-foundedness; and as I noted, you CAN have non-wellfounded sets by simply dropping the Axom of Regularity. Russell's paradox shows that you CAN NOT have unrestricted comprehension. That means that you can always cut down or qualify a set by a predicate; but you can not qualify over the universe. The principle that lets you qualify sets by predicates is the Axiom schema of specification, as I mentioned. Well, nothing but the fact that in your very next sentence you used it exactly that way I think this was in response to my saying back to you exactly what you were using as a definition. But this point may be lost in the cross-quoting. Surely you haven't forgotten that we were talking about ZFC. If you're talking about some other version, please say so. By the way I looked up KFC set theory. There is such a thing. The Google references were VERY obscure and I never found a full definition, but the K stands for Kripke and this is some arcane inside baseball in modal logic. I think this is getting to be a point of semantics. I already noted that there are only countably many predicates but uncountably many sets, so most sets can't be chararized by predicates; and I also gave an explicit construction of a set that can not be characterized by any property of its members. But if you have some vague notion in your mind about this, it's not a point worth arguing. Frankly I was hoping people would find these examples interesting, particularly since I pointed out that the example I gave is the first step in understanding the famous Banach-Tarski paradox. But if I'm not making my point at all, I take responsibility for failing to communicate with sufficient clarity. Oh what a fascinating thing to say. Set theory didn't exist before 1874 (Cantor) and didn't achieve its modern form till 1922, with the publication of the ZFC axioms. Set theory is a historically contingent activity of humans. Now of course you are right that there is a human notion of "collections" that is very ancient (though I'd dispute that it's independent of humans entirely). But at least the idea of collections does predate set theory by thousands of years. Formal set theory is only a recent attempt to formalize our vague ideas about collections. But in the process of formalization, certain naive notions must be abandoned. We have to transition from intuition to doing only what's allowed by the axioms. I think that's the impedance mismatch we're having. If you're using sets for something, naive intuitions are generally ok. But if you're trying to think about the exact nature of sets, then you have to work in the context of formal set theory. And in fact as you note ... set theory really is quite strange. Sets in mathematics are exactly what the axioms say they are; no more and no less. And even then there are all sorts of nonstandard models for even the standard axioms. So yes, set theory is very strange, and the sets of mathematics are not really the collections of our intuition.

If you click the link in your post you'll see that it's not the right link. This is a forum bug. Either that or it only happens for me. Chrome on MacOS FWIW.

IMO the parts that were clear (to me at least) were wrong. The parts about benzene rings and non-wellfounded sets were off point. I'll await further elaboration. But no explicit characterization of a choice function on the equivalence classes of [math]\mathbb{R} / \mathbb{Q}[/math] is possible. That's a mathematical fact. I shall await your response. But there is no explicit characterization of the elements of [math]V[/math]. By the way I did not make that example up. It's a standard idea that shows up in many different guises. These examples are not relevant to the discussion of [math]V[/math], I only mention them for interest. * If you restrict [math]V[/math] to the unit interval and try to apply the standard axioms of probability, you get an example of a nonmeasurable set, a set of reals that can not possibly have a notion of size or probability assigned to it. Tthe traditional use of the letter [math]V[/math] is in honor of Giuseppe Vitali, who cooked up this example in 1905. * If you consider the reals as a vector space over the rationals and then apply the theorem that every vector space has a basis, you get a set of reals [math]H[/math] such that every real number has a unique expression as a finite linear combination [math]r = q_1 h_1 + q_2 h_2 + \dots + q_n h_n[/math] where each [math]q_i[/math] is rational and each [math]h_i \in H[/math]. Once again, the set [math]H[/math] can be shown to exist but is impossible to visualize or characterize via some property. The [math]H[/math] is in honor of Georg_Hamel, and the set [math]H[/math] is known as a Hamel basis. The theorem that every vector space has a basis is logically equivalent to the Axiom of Choice (AC). If you deny AC you then have a vector space that has no basis. Point being that you get weirdness whether you accept or deny AC so we might as well accept it. That's the modern viewpoint. * The same technique, defining some equivalence relation and then invoking the Axiom of Choice to create a set consisting of one element from each equivalence class, is a key step in the famous Banach-Tarski paradox, in which we decompose a sphere in Euclidean 3-space into five pieces, move the pieces around using rigid translations and rotations, and put the pieces together to create two spheres, each the same size as the original. (See step 3 of the proof outline in the linked Wiki page]. I'm just mentioning these examples to show that I didn't pull my idea out of thin air (or my nether regions), but rather out of modern math. Whether modern math itself comes out of thin air is another question altogether. Math really is all about pink unicorns. You're right that I didn't explain equivalence relations, if that's what you mean. I assumed knowledge of equivalence relations and linked the relevant Wiki page. Every equivalence relation partitions a set into a collection of pairwise disjoint equivalence classes. That's a theorem from discrete math class that most people see at some point. https://en.wikipedia.org/wiki/Equivalence_relation. Or did you mean not well motivated in some other way? I agree that there's a conceptual leap to forming sets using the Axiom of Choice. [math]V[/math] is the simplest example I know, but it's deceptively tricky to get one's mind around. I see where you're going and you'll soon be in trouble. Your construction is outlawed by order of Bertrand Russell himself, who in 1901 said: "No you can't do that!!" If we could define a set that way, we could just let [math]P[/math] be the proposition [math]x \notin x[/math]. Then the resulting set both is and isn't a member of itself, a contradiction. This is the famous Russell's paradox, which blew up naive set theory in the early 1900's. [sorry no Wiki link but it can easily be looked up. Forum software doesn't like the link]. The resolution is to disallow exactly the type of unrestricted set formation you attempted. Rather, you have to start with some already existing set, and then apply a predicate (or proposition) to it. So if [math]Y[/math] is some set that already exists, you can take any proposition [math]P[/math] and define [math]X = \{x \in Y : P(x)\}[/math]. Without the restriction to an already existing set, you get a contradiction. In general, we can say nothing at all. That would not suffice as the defining property of the elements of some set in Studiot's sense. It's true that every set is the set of its elements. But you can't use that fact to identify which elements are in the set and which aren't. Indeed, since there are only countably many propositions (a proposition being a finite-length string over some countable alphabet) and there are uncountably many sets (of natural numbers, say), it follows that most sets cannot possibly be uniquely characterized by a property. Most sets are essentially random. In terms of computability theory, which is more familiar to people these days than set theory, most sets can't have their members cranked out by a Turing machine. That's because the set of TMs is countably infinite. [META: The forum software doesn't like the Wiki link to Russell's paradox, which contains a special character sequence for the apostrophe. When I try to correct the link in the editor, the software mangles my post completely. I got tired of re-writing the post over and over and finally just omitted the link. Is this a known bug?]

First you'd have to define the symbol [math]\infty[/math], then we'd know what its properties are. Typically [math]\infty[/math] is used to refer to the conceptual "endpoints" of the real number line in the extended real number system. In that context, the symbol [math]\infty[/math] has definite properties such as, "[math]x < \infty[/math] for any real number [math]x[/math]". The symbol [math]\infty[/math] does not refer to an infinite set these days. It's only used in the very limited context of the extended reals. There are other symbols in use for infinite sets, such as [math]\omega[/math], the first transfinite ordinal; and [math]\aleph_0[/math], the first transfinite cardinal. In each case, a symbol is introduced and defined; and from the definition, we can deduce a laundry list of associated properties. If you put some symbol between curly braces you have to define what the symbol stands for. And once you do that, you're specifying its properties. The easiest way to create a set without specifying a defining property or enumerating its members is to use the Axiom of Choice in the manner I showed in my earlier example. By the way did you know that mathematics is all about pink unicorns, nothing else?

Ah the top left icon. Thank you much.

Now I'm hungry! "KFC does not apply to all sets or even all mathematical sets. The foundation axiom limits KFC sets to those for which an epsilon minimum can be defined. The set of symmetries of a regular hexagon, for instance, has no minimum, though it obeys many of the other axioms." Of course you mean ZFC here. In ZFC we have the kernel of a linear transformation. In KFC we have Colonel Sanders. How do you do quoted text on this forum? I can't figure out this quote window. In your example I believe you are confusing well-foundedness (a set can't contain itself as a member) with having a smallest element in some order. It's true that there is no order on a set of permutations, but they're still well-founded sets. A similar example would be the integers, which have no smallest element: -1, -2, -3, ... But the integers are still a well-founded set. The set of integers does not contain itself as a member. When you say that ZFC does not apply to all mathematical sets, that's only true if you are specifically working in some alternative set theory. All the sets we're talking about, including sets of permutations, are modeled in ZFC and are well-founded. I'm afraid I don't know enough chemistry to respond to your comments about benzene rings. But I did look up the mathematical handling of chirality, and evidently this phenomenon does have a mathematical description. https://en.wikipedia.org/wiki/Chirality_(mathematics)

No accusations intended, I assure you. Just trying to understand the question. I'm the one who's confused and looking for clarity. I'm giving this example as it demonstrates how the Axiom of Choice can be used to produce a set whose members have no characteristic property in common, nor are they enumerated or even known. It's an interesting (to me) broadening of the discussion to give an idea of the strange kinds of sets people use in ZFC. You wrote, "Firstly elements may have other properties than they possess as elements of a particular set. Such properties are irrelevent to their member ship of any particular set." Ok. So V is a set of reals. But if some number x real, that fact gives you no information about whether it's in V. The fact that x is real is an irrelevant property as you described it. The members of V are impossible to identify or recognize. All the elements of V are real, but many reals are not in V. And the ones that are, can not be characterized by any property. If I show you a real number, there's no way for you to tell me whether it's in V or not by any conceivable property. That's what I find interesting about this example. To clarify what some of the equivalence classes of V are, one class is the rationals [math]\mathbb{Q}[/math]. Another is the rationals plus pi, that is every number of the form q + pi where q is rational. Another class is the rationals plus sqrt(2). It's clear (if you think about it) that every class is of the form [math]\mathbb{Q} + x[/math] for some real number x; but to get the unique set of classes you can't use ALL the real numbers for x, since some real numbers give the same class as others. For example [math]\mathbb{Q} + x[/math] and [math]\mathbb{Q} + y[/math] are exactly the same equivalence class just in case x and y differ by a rational. Now V is defined as containing exactly one element of each equivalence class. Of course you are right that all the elements of V are real numbers, but isn't that a general property and not a property these numbers have specifically by virtue of being elements of V? I hope that nothing I'm writing is coming across as argumentative or accusatory in any way at all. I find the sets given by Choice to be quite different in character than the usual sets that are given by explicit means, and interesting to contemplate. A set like V pushes against much of the intuition about sets expressed in this thread. But clearly I'm not explaining it well enough, otherwise you'd agree

Reading through the thread, it's a little confusing whether you are referring to properties of elements or of a set. I can't imagine any specific object in the universe, physical or abstract, having exactly one property. In any event, V is a set containing exactly one member of each equivalence class. The rationals are an equivalence class (the difference of any two rationals is rational) so V contains exactly one rational. But it's impossible to say which rational V contains. The reason I posted this example is that it's a counterexample to your suggestion that the two ways to create a set are to specify a property its members have in common, or to enumerate its members. In the case of V, we have no idea which real numbers it contains nor what their properties are; and there's no property or set of properties that characterizes the elements of V in any way. The existence of V is guaranteed, but we can't name or characterize any of its members.

I hope nobody minds my resurrecting this thread. I wanted to mention that if we work in ZFC, the standard axiom system for modern math, there are sets that satisfy neither of those properties. Consider the set of real numbers; and if x and y are real numbers, define the relation x ~ y if it happens to be the case that x - y is a rational number. For example, (1/3 + pi) ~ pi because (1/3 + pi) - pi = 1/3 is rational. This is readily seen to be an equivalence relation. For all real numbers x, we have x ~ x; also for all x and y we have x ~ y implies y ~ x; and for x, y, and z we have x ~ y and y ~ z implies x ~ z. This equivalence relation partitions the real numbers into a set of mutually disjoint equivalence classes. By the Axiom of Choice (that's the C in ZFC) there is a set, call it V, that consists of exactly one representative from each of the equivalence classes. We know from the rules of set theory that V exists. We can not name any of its specific members; nor is there any property, other than membership in V, that's shared by the elements of V. We can't look at a real number and say whether that number is in V. If you ask, "Is 1/3 in V?" I answer that I don't know. I do know that V contains exactly one rational (why?) but I have no idea which one it is. V is a set whose members are not known, whose members can not be recognized as belonging or not belonging to V, and share no property in common. I should mention that this kind of construction is exactly why some people don't like the Axiom of Choice. However if you reject Choice, you get other unpleasant anomolies. So mathematicians just accept it and get used to using it. What that means metaphysically is a personal choice.

New here. Amateur math fan. I'll take the other side of that one. I dispute the truth of CH. CH is known to be independent of the standard axioms of set theory (ZFC). The only way to prove CH true or false is to cook up a nonstandard model of the axioms and show that in that model, CH is either true or false. That's what Gödel and Cohen did, respectively. It may be the case that CH is true (or false) about the universe of sets, and that we just haven't found the right axiomatization. Another view is that the matter has no truth or falsity, since sets are just a formal abstractions with ultimately no meaning. Or it may be the case that there's some other resolution. But you can't say that CH is true without supplying further context. By the way, both Gödel and Cohen were of the opinion that CH is false. A heuristic argument for that position is by analogy with finite sets. Between the powerset of a 3 element set and the powerset of a 4 element set are sets of many intermediate cardinalities. Why shouldn't the same be true in the infinite case? Of course that's not even remotely any kind of proof; but it does give some understanding of why people think CH may well be false. The Wiki page for Cohen outlines his argument against CH along the same lines. He feels that the powerset operation is so powerful that it must make a great leap in cardinalities, not a one-unit step. He thought that the cardinality of [math]\mathbb{R}[/math] must be an extremely large Aleph.