Sorry for starting a lot of topics in set theory; I think this will be my last. I just want to know what are the basics I need to know about set theory to mathematical analysis.

Is what I have written below enough for set theory in relation to real analysis, and will it ensure that I only work with sets, and not so-called proper classes? And also, is what I write correct or wrong? I know almost nothing about ZF axiomatic theory and have no time to learn it right now, or even learn logic properly.

  1. If we only use the basic set operations, like union, intersection, complement, difference, Cartesian-product, and also the axiom of choice, and only use these operations to create new sets on collections we already know are sets, we will not get any paradoxes or contradictions? (I assume in order to prove all of this you have to study a lot of logic and deep set theory, so I do not need a proof.)

  2. In real analysis we sometimes meet very big sets, like the set of all functions on a given space, etc. The reason these sets are allowed to exist, is because if we look at the definition of a function as a relation, and the definition of a relation as a set, then all these big sets we meet in mathematical analysis can be created using known sets, and the basic operations in (1)? Hence they can exist without giving paradoxes like Russel's Paradox (but again, in order to prove this you probably need to study deep set theory). Is this correct?

  3. The real numbers, and all the Euclidean vector spaces we work with are sets without paradoxes, because they are created by the basic operations described earlier of the natural numbers. And the collection of natural numbers is a set without paradoxes because it is easily defined using the basic set operations in (1)? (In this case you also need the axiom of infinity).

In summary, is the reason they don't encounter set-theory paradoxes in "ordinary mathematics" the fact that all their big sets (like the function spaces) can be shown to be created using the set operations on smaller sets like the natural numbers/real numbers, and it is proven in advanced set theory, that if you follow these rules, you do not get paradoxes?

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    $\begingroup$ If you are beginning real analysis, you will not need to worry about these aspects of set theory. Many of these more complicated concepts are harder to understand without a solid foundation as to why they are problematic. The basic set operations (union, intersection, difference, complement), when combined, yield enough power to handle most of basic real analysis. $\endgroup$ – Emily May 22 '15 at 3:45
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    $\begingroup$ You don't need to study chemistry in order to learn to cook. $\endgroup$ – copper.hat May 22 '15 at 3:50
  • $\begingroup$ @Arkamis I have taken real analysis, but I am thinking about trying to take a course in functional analysis and measure theory aswell. But I have no time to take logic or set theory. So I just want to know in some way why the sets I am reading about are allowed to exist(I know I can't get a very good understanding of this right now.) I see that the operations you mention are enough to handle analysis, but is the reason for this what I wrote(or something along these lines?), that it is proved in advanced set theory that if you create new sets following these rules[cont] $\endgroup$ – user119615 May 22 '15 at 3:50
  • $\begingroup$ then you don't get paradoxes? And that all the big sets they create in analyis are allowed to exist because they are all built by these operations on easier sets like the real numbers? $\endgroup$ – user119615 May 22 '15 at 3:51
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    $\begingroup$ To reinforce this point, consider the following example. I have about 20 different analysis books on my bookshelf, ranging from Stricharz to Rudin to Folland. In almost every book, Chapter 0 or Chapter 1 is "set theory". Each book covers your basic operations that are basically exactly the same things you learned when you first learned of Venn Diagrams. (Maybe Folland introduces $\limsup$ and $\liminf$ of sets, and that's it). These are multi-generational texts; none of them find the need to worry about the subtleties of foundational logic. $\endgroup$ – Emily May 22 '15 at 3:59

I am admittedly no expert in set theory, but what I believe to be true is this:

In answer to your questions-

1) The answer to this is unknown. A famous result (known as Godel's Theorem) says that set theory is not "powerful" enough to prove its own (logical) consistency. This does not preclude proving set theory is consistent using some more "inclusive" system-but this rather "begs the question", as we have just "kicked the problem upstairs", our "larger system" must then itself be shown to be consistent, which is in essence the same difficulty we had before. Indeed, set theory is by no means the smallest system which has this kind of problem, the system of natural numbers (ordinary arithmetic) possesses it as well.

Nevertheless, it is an article of faith of most mathematicians that set theory is indeed consistent-it's just not proven. Some believe it is entirely unprovable. So far, no glaring problems have arisen from our "current version".

2) Yes, and no. The "large" sets you meet in analysis can be constructed, in minute detail, from very humble beginnings, starting with the one set we are certain exists, the empty set. But there are certain logical subtleties involved-some mathematicians (called constructionists, or intuitionists) allow a smaller "tool kit" for creating "valid sets", so for these mathematicians, even the "real numbers" are somewhat different than what is commonly accepted. Even amongst "mainstream" mathematicians, who accept most of Zermelo-Fraenkel set theory, there are those who question the validity of the Axiom of Choice, which is used to construct certain "large" sets (such as Vitali sets). The famous Banach-Tarski paradox arises if you accept the Axiom of Choice (not a "true" paradox, no logical contradiction arises, but what Banach-Tarski says is quite counter-intuitve: we can decompose a ball into two disjoint subsets, each of which is a ball congruent to the original).

But these "large sets" you speak of can be constructed, at any rate, using ZF+C (the Zermelo-Fraenkel axioms along with the Axiom of Choice), and no contradictions have yet arisen. This is somewhat reassuring, but it is too soon to say that no possible contradiction can exist.

3) The answer to this is, I'm afraid, rather similar to those above-there is no proof that even the natural numbers are paradox-free. I'm really terribly sorry about this, as I'm sure you would rather hear some other answer.And if even the "simple" natural numbers are not provably consistent, a more involved set such as the real numbers, or the set of all real-valued functions defined on some subset of the reals is likewise not provably "paradox-free".

In fact, I would go so far as to say there is NO certain knowledge in mathematics at all! Rather, all our mathematical knowledge is contingent, and some (possibly unfounded) assumptions must be made, and the only things we can be sure of, is what consequences our assumptions have. This is nothing new: Euclid's Elements starts with Postulates; a modern mathematician is more likely to use axioms. Certain axioms may have a certain intuitive appeal to you, but that is a far cry from a demonstration of "self-evident".

Axioms are chosen, for lack of a better term, on how "appropriate" they are: when we look at a problem, we seek to model it, to hopefully expose unobvious aspects. For example, when counting sheep, natural numbers seems like a "good fit", and might allow us to discover if a particular sheep has been eaten by wolves (or otherwise disappeared), but it is clear this model does not account for the individual differences each sheep has. My point is, it is still a matter of judgment which mathematical methods are most appropriate for solving "real" problems, and although calculus (for example) has proven quite successful in the physical sciences, that does not mean the qualities of real objects (mass, density, luminosity, etc.) are in point of fact, actually real numbers, no matter how suggestive the name.

It seems rather curious that mathematics, an undeniably rational endeavor, should require faith at its foundations, but that appears to be the case.

  • $\begingroup$ Thank you veru much for writing such a long answer. I just have some smaller questions. I see that you mention Godel, and his theorem makes this difficult. From what I understand he proved that the theories can never prove themselves etc., and that is why we don't know if we can have paradoxes etc., in the points I mentioned? But even still, there must be some distinction between the fact that mathematicians know that some paradoxes may or may not occur, and the fact that we know that paradoxes such as Russel's paradox exist? Another way of asking my question is maybe then: we can never[cont] $\endgroup$ – user119615 May 22 '15 at 12:43
  • $\begingroup$ know if a theory is paradox free or not, because of Godel. But if we always follow the rules of union, intersection etc.., when creating new sets, the results of what we get are not known to have any paradoxes? We can't be certain, because a theory as you say can not prove that it is paradox free, but if we follow these rules, there is atleast not shown that it leads to a paradox? Is this the reason that they in real analysis can work with these complicated sets? We can never know that what we are working with does not contain a paradox(Godel), but if they follow the rules of [cont] $\endgroup$ – user119615 May 22 '15 at 12:47
  • $\begingroup$ constuction new sets, they atleast not have known paradoxes, like Russel's paradox? $\endgroup$ – user119615 May 22 '15 at 12:48
  • $\begingroup$ Simply stated: By following the rules of creating new sets(union, intersection..), we can never know if we will end up with a paradox or not(Godel), but by following these rules we do not end up with known paradoxes like Russel?, and that is why the complicated sets in the real analysis text book work?, all the sets are created from sets we know not involves any known paradoxes?, so we believe we are ok?, but we can never be sure about unknown paradoxes, beaucse a theory can never prove that it is paradox free(Godel)? $\endgroup$ – user119615 May 22 '15 at 12:59
  • $\begingroup$ The rules as we have them now were created in part, specifically to avoid Russell's paradox. Certainly with finite sets, everything seems to behave as expected. The first versions of set theory wanted to say something like this: "a set is completely determined by a (any) property that determines which elements it has". The problem with this is that some possible properties are self-negating-the classic example being: "this sentence is false". The fix to this was only allowing "comprehension" over already existing sets, which amounts to building up a family of sets in the ways you describe,. $\endgroup$ – David Wheeler May 22 '15 at 22:50

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