Find solution
Is there a natural topology on the set of open sets?

Why is the concept of the set of all sets contradictory?

  • I read that the set of all sets is a contradictory concept. I thought about this. If U is such set, then U must be an element of U (not only that U is - of course- a subset of U, but that U belongs to U like 2 belongs the reals). Though this sounds kinda weird to me, I'm not sure if this means a contradiction. Then I noticed that every element of P(U), the set of parts of U, must be in U, so that p(U) ⊂ U and, therefore cardinality(U) ≥ cardinality(P(U)). But we know that, for any set A, cardinality(A) < cardinality(P(A), so that now we have a contradiction. Is my reasoning correct? So, though this is apparently possible (after all, we can define sets of anything) there's no such thing as a "set of all sets"?

  • Answer:

    One needs to be careful with words here: let's say that a "set", properly defined, is a collection of objects that may be considered a single object. We may now ask the question: are all collections of objects "sets" in this sense? The answer is no: there are collections that are not sets; they are, in a sense, too large for that, and are called "proper classes" (the name cames from the fact that every set is a class, but the reverse is not true). One of such collections is precisely the "(proper) class (not set!) of all sets". It is not a contraditory concept in itself; as I pointed out to you in another answer, there are formal theories of sets (ZFC being the most well-studied), but there are also formal theories of classes (the most common is the NBG, from Von Neumann-Bernays-Gödel, theory, but they're not widely used, or even known, in Mathematics outside Mathematical Logic and Set Theory, simply because almost all Mathematics is done in "small" sets), which contain Set Theory as a fragment (or subtheory). In fact, the above class appears in Set Theory (not as a set), and is usually called the "Von Neumann Universe", or the "Cumulative Hierarchy", and is denoted by V. Why then is then the existence of V sometimes considered contradictory? Because if you consider it a set, then contradictions do appear; in fact, you gave one yourself, when you assigned a cardinality to it, something that can be done only with sets. Other, more commonly known, contradictions, arise from subclasses of V. The most well-known is (as pointed out by ksoileau), probably, the Russell class Ru: the class of all sets that are not members of themselves. More compactly: Ru = {x: x ∉ x} (x is a variable that ranges over sets) Now, if you treat Ru as a set, then you may ask if Ru ∈ Ru, but, if yes, then Ru ∉ Ru (by definition of Ru). If no, then Ru ∈ Ru, and there's no way of escaping this (if you persist in considering Ru a set; as a proper it's a perfectly legitimate object). Regarding your last remark about existence, the answer is that nobody knows (but bear in mind that your assertion that we may define sets of anything must be qualified, not only by what is said above, regarding the distinction between sets and proper classes, but also because other technical and philosophical problems, regarding something called "urelements": set elements that cannot be defined in terms of other sets, but they're not relevant here): it's not a mathematical question, but a philosophical one. There are mathematicians (and philosophers with an interest in Mathematics) that do not even believe that the larger sets in V exist, and there are others that advogue the introduction of additional axioms in ZFC that would allow even larger sets into the theory. It's a minority debate, but it goes on, and there is no end in sight (and not only for these more exotic objects: the questions regarding the ontological status of mathematical objects are the cause of many wars between philosophers).

Taty at Yahoo! Answers Visit the source

Was this solution helpful to you?

Other answers

I don't see how the concept of the set of all sets is contradictory, but I'm not an expert, so someone else might be able to cast light on that matter. However, the operation of forming sets must be done carefully to avoid contradiction. For example, here's Russell's paradox: Define T=the set of all sets which do not contain themselves as a member. Does T contain itself? If it did, this would disqualify T for membership in itself. Therefore T does not contain itself. But this qualifies it for membership in itself. So T contains itself. Contradiction.

ksoileau

JCS is right, but I just wanted to add a couple points. Note that in the axioms of ZFC set theory, there is no assumption that such a "universal set" exists, and in fact, its existence is, as JCS mentioned, outside the question of ZFC set theory itself. The problem arises more explicitly if you think about it like this: a set is a collection of objects. Then we can define a "superset" as a collection of sets. Then we can define a "supersuperset" as a collection of supersets. Do you think, continuing on like this, we will ever arrive at a "universal set"? What's interesting about this approach is you immediately realize standard set-theoretic constructions, like unions and ordinals, are of little use. This is somewhat related to the concept of an inaccesible cardinal, but is not exactly the same thing. Also, note that the axiom of regularity in ZFC precludes the existence of such a set U where U belongs to U. Steve EDIT - I just wanted to clarify what I meant when I said "outside the question of ZFC set theory itself." This is a mathematician's point of view - I don't care if ZFC exists in some universe or not, I take it as axiomatically true. There are groups of "realists" who argue such a "universal set" provides the intuitive motivation behind the axioms of ZFC, and there are others who insist such a universe must exist. This is a philosophical question, however, not a mathematical one (if such a fine line exists).

Related Q & A:

Just Added Q & A:

Find solution

For every problem there is a solution! Proved by Solucija.

  • Got an issue and looking for advice?

  • Ask Solucija to search every corner of the Web for help.

  • Get workable solutions and helpful tips in a moment.

Just ask Solucija about an issue you face and immediately get a list of ready solutions, answers and tips from other Internet users. We always provide the most suitable and complete answer to your question at the top, along with a few good alternatives below.