Russell's paradox Puns

In 1901, mathematician and philosopher Bertrand Russell was investigating set theory, a formal way of defining and dealing with collections of anything. At the time, one of its central ideas was that for every property you can define, there must be a set. There’s the set of all green things, and the set of all whole numbers except 4. You can also define sets of sets: say, the set of all sets that contain exactly two elements. The problem comes when pondering the possibility of a set of all sets that do not contain themselves — this seems to be impossible.

The paradox exposed contradictions in much of the mathematics of the time, forcing Russell and others to try to devise more intricate logical footings for mathematics. Russell’s approach was to say that mathematical objects fall into a hierarchy of different “types”, each one built only from objects of lower type. Type theory has been used to design computer programming languages that reduce the chance of creating bugs. But it’s not the definitive solution

Update: You can't address something that you aren't aware of, but you also can't correct definitions that are already consistent... So it looks like Cantor knew that there could be problems with allowing any collection of objects to be a set, but he certainly was not aware of what those problems could be. Frege, on the other hand (from what I read), stood behind his Basic Law V (naive comprehension), which states (in my own words - correct me if I'm wrong) that, given a property, you can create a set of all sets that each satisfy that property. This means that Frege was incorrect by believing Basic Law V, and Cantor was not inconsistent by being conservative in his definitions and by requiring that collections be well-defined from the get-go. I have no evidence of Cantor believing Basic Law V (I didn't look that hard into this though), and because of that Cantor was never wrong about his set theory - he just wasn't aware of how to formalize it. Neither of them knew what problems existed with Basic Law V nor were they aware of how to address things that had not existed yet. I have more questions that I will be asking on this sub. Thanks everyone for helping me understand this history.

Original question:

I translated one of Cantor's articles, and saw that he defined a set to be ~ a collection of well-defined objects ~ which does not seem to logically imply Russell's paradox, considering the notion well-defined can require ZFC-axioms, and so on, right? Cantor doesn't go on to explain what well-defined really means (at least as far as I read), but that's not really the point. There's a problem when you say that sets are collections of objects, since there are collections of objects that should definitely not be sets. I don't see a problem when you add the requirement that the collection is well-defined, since this removes ambiguity in whatever way you want, depending on how you want to remove ambiguity.

I notice textbooks use universes to try to solve the problem of naive set theory in an attempt to kick the philosophical can down the road into courses that are more suitable for discussion on ZFC-axioms, and Cantor's use of well-defined seems very analogous to that solution. Am I missing something? Was Cantor really the author of naive set theory? Was Russell solving a paradox that was already kind-of solved by Cantor here? Did Cantor know why he needed to use that wording well-defined? This makes me feel like naive set theory was never

I worked on this project for a few weeks, and tried synthesizing lots of concepts in set theory. Any feedback on how these videos can be improved would be great. All content is released under Creative Commons "No Rights Reserved", so please feel free to use this video however you want. Thanks to Nathanfenner for taking the time to comment on my last video. I hope this video is more productive for everyone.

https://youtu.be/UjklRuyFaIA

My prof. said that in the postscript or appendix to Frege's work, Frege commented on Russell's paradox being a real problem.

I want to put an excerpt of that comment in my logic notes because that presents the topic with a sense of history which is good, I think.

Does anyone know the quote? (Otherwise, my notes will be without charm unless I go to the library to find the quote myself, but I think I have an overdue book.)

Thank you

Hi all,

I've been listening to a lecture by Alexandre Stevens where he says that there are only two logics of sexuation because there are only two possible responses to Russell's paradox.

Stevens talked about the positing of the exception on the male side, but I can't quite grasp the technical execution of this specifically as a response to Russell's paradox, let alone the female side.

Can anyone help me think this through?

It has to do with the train of thought I was pursuing in my previous post on topology, so would be interested to hear from logicians who can shed light on this problem in a topological register.

I have a vague grasp of Russell's paradox and I was curious whether it has to do with more than just set theory? I did a fair amount of google-ing but I couldn't find anything besides that, if anyone has further information or resources it would be greatly appreciated!

The Stanford Encyclopedia of Philosophy's says Russell's Paradox does not arise under a paraconsistent logic without contraction (see here). However, I remember reading somewhere that just removing contraction is enough to avoid the paradox, thus affine logic (linear logic with weakening) should suffice. Is that so? If not, how to derive the paradox within affine logic?

Welcome to Saint-Louis! It is at this presigious institution that an incredible new result has been proven, the likes of which will make things like the pesky Axiom of Regularity (EDIT: The Axiom of Regularity is not what was used to solve Russell's Paradox. The Axiom schema of Specification is what solved the issue) entirely redundant. If only set theorists had spent more time thinking about dogs!

> What exactly is it even supposed to mean for a class to be a ‘member of itself’? It may sound similar to being ‘identical with oneself’ but they are not equivalent. Being identical with oneself is tautological; that is true of everything. But a category, class, or set could never be a member of itself. That claim does not even make sense; it would be like saying that all of the cells in my body are also a cell of my body.

A very bold claim that really makes you think, huh? What is the difference between membership and equality? I personally can’t think of anything.

> If the category ‘dogs’ was a member of itself, what would its other members be, exactly? I assume that there would have to be other members, in addition to itself, or we would just say that it is one category. So what would these other members be? Are they dogs? If they are, why would they not be included in the member-class? If they are included, and both the superclass and the member-class have exactly the same members, and exclude exactly the same members, then what is the distinction between them that would make them two separate categories?

I think this is an obvious result: if x ∈ A and A ∈ A, then x ∉ A. Also A = A and A ≠ A? I think it is important to note that the author seems to be very confused about the difference between set membership and set containment. In fact, I’m pretty sure they've read nothing but the Wikipedia page for Russell’s Paradox and then declared themselves an expert on the topic. The next few paragraphs are just the author repeating themselves but with shittier examples, so here are a few highlights:

> ‘Ice’ is not identical to ‘water’; ‘water’ is not a species of itself.

> ‘All things identical to Socrates’ would have an infinite number of members because the superclass must include all the members of its subclasses. (‘Mammals’ must include all members of ‘dogs’, ‘cats’, ‘horses’, etc.)

>A class is a collection of things, not a ‘thing’ itself. [Russell] claims that ‘“everything” is something’ but a

I found in this passage from Ray Monk's biography of Wittgenstein (p.307):

"He now wanted to declare such contradictions trivial, to declare that, once the fog had cleared and these sorts of problem had lost their nimbus, it could be seen that the real problem was not the contradictions themselves but the imperfect vision that made them look like important and interesting dilemmas.You set up a game and discover that two rules can in certain cases contradict one another. So what? 'What do we do in such a case? Very simple - we introduce a new rule and the conflict is resolved?' "

I've looked a bit in Philosophical Investigations but I can''t find it there. Maybe this particular remark is from something unpublished but I would be grateful if anyone could point me in the direction of anywhere that the later Wittgenstein discusses this sort of thing.

I think it was a stick figure comic in the style of XKCD. A suspect who is a mathematician is being interrogated, and the two cops are playing Good Cop-Bad Cop.

The Bad Cop holds up a sheet of paper with two circles. One says "The set of all sets that are members of themselves" the other circle says "The set of all sets that are not members of themselves"

The Bad Cop says "you like these two? How about I move them a little closer together?" which is basically torturing the mathematician by referring to Russell's paradox.

Come to think of it, it could have appeared in the University of Waterloo's mathNEWS around 1999~2004.

So Russell's paradox is a contradiction in naive set theory where if we define R = {x | x ∉ x} then R ∈ R if and only if R ∉ R.

There is an algorithm which is similar to R. Let Q be an algorithm that takes one input M and does the following: Interpret M as an algorithm, and execute that algorithm on its own description. If it accepts, reject. If it rejects, accept.

For example, Q would accept the algorithm which rejects everything (analogous to the empty set), and reject the algorithm that rejects everything (analogous to the universal set). What about Q applied to itself? Well, it simply would not halt, neither accepting nor rejecting.

This leads to the idea of a "theory of computable sets", where for two algorithms X and Y, we say X ∈ Y if Y accepts on X, X ∉ Y if Y rejects X, and say X ⋔ Y if Y does not halt given input X. Note that X ∉ Y would no longer be the negation of X ∈ Y in this theory. Rather, for any two sets X and Y, we know at exactly one of the three above cases is true.

This set theory is consistent since it has a model, avoiding all the set theoretic paradoxes. One caveat is that ⋔ can not be used in set comprehension. However, it is quite limited; it only captures computable phenomenon, so to speak. We also extensionality, and it is not clear who to restore it. However, I wonder if there is some way to generalize or modify this concept to create a powerful set theory.

One way would be to extend what we mean by "algorithm" in to allow it to carry out more mathematical operations. Algorithms in this sense would no longer correspond to computations, but they could share certain properties with computations allowing the set theory to still be consistent. One obvious method is just to use turing machines with absurdly powerful oracles. For example, if we use 0^((𝜔)^) as the oracle, any arithmetical set could be defined. With a stronger oracle we could go up to implicitly arithmetical sets. However, going "too far" could cause problems for axiomization, since we do not have a background set theory to work with. This may lead to a limitation on what sets we can define, or too many statements being independent of the theory. (Oh, and you may be wondering how we could possibly axiomatize 0^((𝜔)^) since it is uncomputable. The answer is that the axioms would only describe it, not compute it. After all, ZFC can describe 0^((𝜔)^) in great detail without computing it. There are many statements

Can anyone give me a better/simpler explanation of Russell’s paradox? I have read a lot on it but I can’t seem to grasp the paradox and the language that is used to explain it conventionally

While reading through "Naive Set Theory" by Halmos, I stumbled upon the famous paradox. As Halmos ominously stated, this means that nothing contains everything, and that, hence, there is no universe.

Isn't this paradox easily 'fixed' by initially defining that x ϵ' x for all x?

I mean, given that x ϵ' x for all x, if A is an arbitrary set, and B = { x : x ϵ A, x ϵ' x } then B has exactly the same elements as A, which, by the axiom of extension, means that B=A. And then again this avoids the paradox, as B ϵ' A because A ϵ' A.

And because everything is a subset of itself, A could conceivably be the set of all sets. Because it would contain itself.

There is an universe! Quod erat demonstrandum!

But what do I know? Is the definition that x ϵ' x for any x problematic or not consistent?

Oh, and as an extra enquiry: what can I read after Halmos? I've found set theory to be really awesome

Discussing with a philosopher about impossible things existing or being allowed within a particular logic system, he told me:

"This is a funny thing about logically impossible things. You can prove that they exist in any non-consistent or paraconsistent logic system. You might even be able to give a (nonsensical) description that satisfies some specific definition of said thing. But that still doesn't give you anything that makes sense. There, you are asking not only to prove that a very particular impossible thing exists, but you are asking for a detailed description of it to exist as well. I know of no method for doing that" (talking mainly about a solution that makes sense to Russell's set paradox)

So ia it there any method/logic system or anything else where impossible/illogical/inconsistent things would be allowed? For example, If a solution to Russell's set paradox cannot exist and it is impossible to exist, is there any method/logic system or anything else where this solution could exist?

OM

Could someone explain to me how Wittgenstein deals with Russell's paradox (and at the same time explain his criticism of Russell's 'type theory') please? I can't seem to grasp the significance of his argument.

The Stanford Encyclopedia of Philosophy's says Russell's Paradox does not arise under a paraconsistent logic without contraction (see here). However, I remember reading somewhere that just removing contraction is enough to avoid the paradox, thus affine logic (linear logic with weakening) should suffice. Is that so? If not, how to derive the paradox within affine logic?