I disagree, obviously.

The OP clearly did exactly what I wanted it to do. It was chosen for its value as a center of conversation, that was the defining reason.

The only thing you have really expressed is your disproval of me, then you made some lame subjective argument of how you think I should have approached this. Your opinion has been noted, as an opinion.

You should. Context is important — Akanthinos

I get the context I need to understand the underlying concepts and do the math. Honestly do you have any idea how many theories and theorems a student of mathematics has to cover? You are not suggesting a realistic approach. I get a brief history, a full proof and than we go. All I really care about personally is the proof and the notation. As long as I can follow proofs and read notation nothing is out of scope.

I mean, some of us will have had around 40+ hours of courses — Akanthinos

I have easily spent more than 40 hours on set theory. It's math and if you speak the lingo I'll understand it. I reviewed all the notation and the underlying concepts for this paradox before posting, I knew before hand the axiom schema of specification would be the main counter point. I also know that several of the arguments presented in this thread don't hack it.

Your lording is not convincing me.

No, rather, you have been missing the context ever since you started this thread. You cannot ask (at least, meaningfully) someone to resolve Russell's Barber if that person hasn't shown an acceptation of unrestricted comprehension. — Akanthinos

I am fairly sure I did, so clearly I can.

Btw, do you know what ad hominem is?

I study math, and I pay very little attention as to where it came from. However, as I always do, I spent time studying this paradox before posting it. I was well aware of the proposed resolution.

He was talking about the layperson, you are missing the context. Also set theory exist in mathematics, I am able and trained in reading the notation and employing sets in formal proofs.

We don't need to reinvent the wheel with every thread.

Let me explain the purpose of these conundrums that I have been posting.

They are here to generate discussion and I don't care if they are paradoxes, contradictions or whatever, just as long as they are meaty enough to get a fruitful discussion going. I also don't care if the side I have aligned myself with is right; I hate it when everyone agrees with one another, it is the most unproductive form of discussion. The purpose of this is to walk away with just a bit more understanding and the paradox is merely the raft we use to cross those waters.

We are at seven pages here now, so I think this one was a good one.

Wait, let me get the dead horse for you. . . .

Call it whatever you like, but changing the name is not a real argument.

As I've said elsewhere, the reaction of the average layperson will almost certainly be, "Oh, it's a trick." In my part of the world it might be worse: "I always figured math was bullshit -- guess I was right." — Srap Tasmaner

I shared this paradox with lots of people, non-math people, not a single one of them reacted that way. I share every paradox I post here with friends and co-workers, it acts as a short conversation piece sometimes; some find them interesting, most don't really care.

Maybe you are underestimating the "average layperson".

I understand that some feel the axiom schema of specification resolves the paradox, I was speaking generally to the notion that if it is nonsense we can ignore it. As that argument seems to repeat itself in each of my threads. Mainly I just wanted a thoughtful reply instead of just a brush off and a straw-man. We can't just dismiss mathematical paradoxes as nonsense, that is not a valid solution.

You can't make mathematical paradoxes go way by just claiming they are nonsense, as even if they are nonsense, they arose out of our math. This is not just the happenstance of two contradictory terms placed in the same sentence, they are rooted in mathematical theory. This means either our math is fundamentally wrong or it is incomplete.

Well it is right in that post you quoted, " it still exist if you apply proof by contradiction to the question."

The fact that it can be shown by existing foundations is what sets it apart.

Also, and this is very important, straw-mans are in no way a proof. Russell's paradox is concerning set theory, saying things like there can't be a round square is just babbling unrelated observations.

You just described one half of the paradox. We already know it can't be resolved, that is why it is a paradox.

Also just drop the barber example, as you are falling off track.

If your solution is the paradox does not exist because the paradox has no solution, then all you are doing is describing a paradox, not resolving one.

An empty set is still a set.

Russell's Paradox remains unresolved and it seems that the only approach that has been suggested so far, is to pretend it is not there.

You have not given any argument as to why it must be denied.

People focus on the barber, as they tend to believe he is the center of the paradox. Maybe this is becuse he is the subject of the sentence, and for that draws all the attention; however, the barber is not the center of the paradox. The paradox is whether he belongs to the set of men who get shaved. In this sense, it is exactly Russell's paradox.

Sure it is a good prior, I don't disagree with that; however, a sample over many coin flip would actually give you a distribution of loaded flips. Which we could use to estimate the population mean, which is still loaded. This is important, as we need to remember this is not truth. When people mix up truth and science, it stops being objective.

I don't think that real fair randomness actually exist. Maybe you view that as trivial, but I don't.

Certainly different than claiming he cannot exist at all and if he could carry it out then it would not be a paradox.

Actually all coins are loaded, no coin flip is exactly 1:1. Fair coin flips only exist in our imagination or possibly also computer simulations, I don't know computers well enough to say. Otherwise we are working on a good enough to be called fair basis, but then that is nearly all of mathematics.

The truth is all of math and all of science is a lie based on inaccurate assumptions, but they are still the best tools we have.

The barber is a simple example that is it; one that I think reads clearly as Russell's Paradox, but if a reader disagree then fine. It is only an example and I don't see how it can be productive to endlessly discuss its merits as an example. The paradox itself, as it is classically defined, is still in the main body of the the OP and is completely independent from the barber example.

Say this barber was a real person, and this is the task he defined to set out to do. Then when he got down to trying to decide if he should shave himself or not, based on his predetermined conditions, would that make him pop out of existence? No, of course not. For the record.

Proof by contradiction does not actually resolve this paradox, it still exist if you apply proof by contradiction to the question: Is R a member of itself? In fact it only proves the paradoxical nature of the question under the assumption: Let R be the set of all sets that are not members of themselves

Also, in neither math or logic are straw-mans valid methods of proof.

It makes complete sense and I am not interested in your straw-man.

https://thephilosophyforum.com/discussion/comment/183352

I actually already addressed this argument of yours.

I need a good explanation of why you think that is relevant before I spend my time on it. I am not trying to evade or be rude, I just don't have a lot of free time and I need to hear a good justification of the parallels.

I don't think you actually know anything about statistics. I think you are confusing probability, a branch of mathematics with the branch of science known as statistics. Which, actually is an understandable mistake; however, the conceptual aspects of statistics is largely contained in the science.

You can't resolve a paradox but simply stating that it is not a paradox. A paradox by any other name is still a conundrum.

Every single one of these threads I have made someone jumps out and goes, "Oh it is not a paradox, therefore paradox resolved." It gets old and I get tired of going back and forth on that point. I mean it is actually moot whether it is officially a paradox or not, the conundrum doesn't fade away just because someone decided not to call it a paradox. So it is easier just to tell people it widely recognized as a paradox, or something along those lines and I am not lying, these are well known paradoxes.

So call it an appeal to authority if you like. I don't really think it falls as neatly in those lines as you do, but either way it is an effective approach to move the discussion off a moot line of discussion.

Also you forgot the link to the OED, which provided the definition of a paradox. Try reading it, as it turns out contradictions can be paradoxes. Why did I use the OED? Because it is an authority. Appeal to authority is not a reason to shrug off a valid authority.

I may have been off somewhat on appeal to authority, but that is not a reason to sweep aside 100 plus years of history, especially when we are talking about language as how people use language for over 100 years is how language is defined. If people have been calling it a paradox for over 100 years, guess what, it is a paradox.

I don't accept your authority on the argument of authority, guess that means you are wrong.

Argument from authority is only wrong if the authority is misplaced. I think over 100 years of history is a very strong authority.

Claim: We cannot form sets out of arbitrary predicates.

Proof:

Assume the negation of our claim: That is, assume that we can always form a set out of a predicate.

Consider the predicate P(x) = "x ∉ x".

Now we let R be the set R = {x : P(x)}. We see (following Russell) that we must have both R ∈ R and R ∉ R. That's a contradiction. — fishfry

From the OP:

Let R be the set of all sets that are not members of themselves

Is R a member of itself? If so, then it must meet the condition of not being a member of itself, which would mean it is not. If it is not, then it must meet the condition of not being a member of itself, which would mean it is a member of itself. — Jeremiah

The question is "Is R a member of itself?"

There are to possible statements we can derive from that question. Our statements to be proven would be: R is not a member of itself and R is a member of itself.

Proof by contradiction would lead us right back to Russell's Paradox.

It seems you have another contradiction on your hands.

Where is yours?

I can tell that writing is not your strong point.

The barber shaves P and Q

The barber shaves Q.

Those are two different things.

At the very least your choice of wording is very ambiguous.