That's is literally what I said. My post:

I'm saying that even as metaphysical dialetheist I do not believe a "barber who shaves all and only those who do not shave themselves" cant exist.
— Me — MindForged

Are you serious? Your OP:

The town barber, who is a man, shaves exactly every man in the town who does not shave himself.
— Jeremiah — MindForged

You added an "and"? It is not "all and"; he shaves " every man in the town who does not shave himself." He does not shave people who shave themselves. That is what creates Russell's Paradox.

There is no semantic difference between what I said and what you said. "All and only" will capture the same set of objects as "every".

There is a huge semantic difference. You are talking about two groups the OP is talking about one.

No. Russell's Paradox is about the set of all sets that do not contain themselves. It does not mention barbers or shaving.

The barber paradox splits a town into two distinct groups: those who shaves themselves and those who don't. The barber is proposed to be the one who shaves those in the latter group. The paradox comes from the fact that the barber must end up being part of both groups. There's no paradox without 2 groups.

Break it into symbols and you'll have Russell's Paradox. But I need to go to bed.

Except you had the barber shaving two groups.

That's where the paradox comes from. He ends up having to shave himself and shave those who cannot shave themselves. He can't do both.

I find it very interesting you are now calling it a paradox. You realized your mistake.

No. We won't.

Can't? Don't know how?

No

Worried you can't read?

No

Well at this time, that is my conclusion.

??? Bro, I said it was a paradox on the previous page. I accept Russell's Paradox but I'm an outlier. The Barber Paradox is simple: no such barber exists, such a barber is fictional.

Let's cut the game, when you added the word and you changed the sentence.

Dude, I "changed it" A) By mistake, since I gave the classic wording of the paradox and B) the words I changed didn't alter the paradox at all. "All and only" is the same thing as "every".

This is ridiculous.

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.

I didn't say the barber shaves two distinct groups P & Q. I said the barber "shaves all and only those who do not shave themselves". That is a condition defining one set.

I can tell that writing is not your strong point.

How's everything going with your contributors?

As I say, that's a somewhat naive view. The specification scheme allows one to avoid the paradox, but it doesn't necessarily solve the paradox. The whole point of regimenting set theory this way was to make make math consistent (or at least not provably inconsistent). But it comes with well known issues, like a number of unsolved questions that have known answers in other systems (e.g. Continuum Hypothesis). — MindForged

MindForged, you are completely misunderstanding the difference between a veridical paradox and a plain old proof by contraction. Moreover, Russell's paradox has absolutely nothing to do with Gödelean incompleteness. Simply nothing.

Let's start by reviewing how proof by contradiction works. I'll use Euclid's classic proof of the infinititude of primes.

Claim: There is no largest prime.

Proof:

Assume the negation of our claim: Assume to the contrary that there is a largest prime.

Then we may number the primes p1, p2, ..., pn. [Moderators: It's so easy to add MathJax to a website. Pretty please? It would greatly enhance mathematical discussions here].

Form the number P = (p1 x p2 x ... x pn) + 1.

Clearly P is not divisible by p1, it leaves remainder 1. Likewise P is not divisible by any of p2, p3, ..., or pn.

Therefore P must either be prime; or else it must be divisible by some prime other than the ones we listed.

Therefore since the assumption that pn is the largest prime leads to a contradiction, we must conclude that in fact there is no largest prime.

Ok, that's the basic pattern. Now observe that Russell's "paradox" follows the exact same form.

Note that calling something a paradox doesn't make it a paradox. Naming conventions are generally historical accidents. For example the Axiom of Choice, Zorn's lemma, and the well-ordering theorem are respectively an axiom, a lemma, and a theorem. However they are all logically equivalent. The names are just accidental conventions. If we called it "Russell's theorem," or "Russell's smackdown of Frege," much less confusion would ensue. I hope this point is clear. From now on I'll refer to Russell's argument as Russell's smackdown.

By the way Frege is credited as being the inventor of the universal and existential quantification operators. He was really quite a bright fellow, notwithstanding his public humiliation at the hands of Russell in this particular matter.


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.

Therefore we conclude that our assumption is false; and that we may not arbitrarily form sets out of predicates.

Now we see that Russell's smackdown is nothing more than a traditional proof by contradiction; a basic pattern of logical reasoning that goes back at least two millennia.

How you get from this to invoking Gödel's incompleteness theorem I simply don't see.

Nor do I see how Russell's smackdown is a genuine paradox. After all, one could argue about primes as follows:

Primes get increasing rare as numbers get big. The farther out you go, the more distance there is on average between primes. If you go out far enough, it's reasonable that you simply run out of primes; and that from some point onward, every number is composite.

If one has this intuition, one would regard Euclid's beautiful proof as a veridical paradox. But very few people would call the infinitude of primes a paradox. Rather, it's a mathematical fact that's often proved via the method of contradiction. If one said, "But I REALLY have this intuition that there must be a largest prime, so Euclid's proof is a paradox that must be resolved," they might indeed have strong feelings about the matter, but their point would not get any traction among mathematicians or even logicians.

Likewise, Russell's smackdown shows that our naive intuition about sets -- that they arise from predicates -- is wrong. Perhaps one has this intuition; but with a little mathematical training, one quickly realizes that to form a set we must apply a predicate to an existing set. That's the axiom (schema) of specification.

Let's see how this works in practice. If N is the natural numbers, what is the set R = {x ∈ N : x ∉ x}? Well, is 0 ∈ 0? No, so 0 is in R. Is 1 ∈ 1? No, so 1 is in R, Continuing in this manner we see that in fact R = N. The axiom of specification has completely resolved the matter. We CAN'T necessarily form sets out of predicates, but we CAN form a set by applying a predicate to an existing set. Done and done.

So when one says, "My intuition is that the primes get so rare that eventually there's a largest one and then no more after that," or "My intuiution is that you can always form a set from a predicate," these are indeed intuitions that an untrained person might have. But with a little mathematical training, one comes to understand and internalize that there are infinitely many primes, and that in order to form a set from a predicate, one must first start with a known set.

This has NOTHING to do with Gödel, nor is the Continuum hypothesis even remotely, by any stretch of the imagination, the same type of phenomenon.

MindForged, you are completely misunderstanding the difference between a veridical paradox and a plain old proof by contraction. Moreover, Russell's paradox has absolutely nothing to do with Gödelean incompleteness. Simply nothing. — fishfry

I didn't mistake anything. The paradox resulted from rules that Frege thought were indubitable. And you completely misrepresented what I said. I didn't say RP had anything whatsoever to with the Incompleteness Theorems. I said the regimentation of set theory was motivated by a desire to.prove mathematics was consistent and complete, but Gödel showed you could only have one or the other, not both (if your system is capable of articulating arithmetic truths).

I know how a proof by contradiction works, pleasedon't patronize me. What you don't seem to get is that avoiding a paradox is not the same as proving your solution ought to be adopted. I can avoid the Liar paradox by disallowing all self reference. But then all hell breaks lose because perfectly sensible sentences like "This is an English sentence" get the axe. The solution has to be justified. Modern.maths avoided the paradox to get on with the business of mathematics, no one disputes that the resolution was ad hoc. They simply take that theoretical black mark as preferable to an inconsistent theory. If the principles which cause the paradox are more theoretically virtuous than the consequences of evading the paradox, then one ought to accept the paradoxical conclusion. I mean, to just throw up "It's a proof by contradiction" is easily abusable:

Early calculus was indisputably contradictory. One had to treat infinitesimals as a non-zero value at one step of the proof and then as having a value of zero at another step. Ergo, we should have dropped calculus as a viable mathematical branch immediately instead of it knowingly being left to languish as an inconsistent theory for 150+ years.

Long story short, it's a lot more complicated than you seem to think. Paired with a Paraconsistent Logic, one can work with Naive Set Theory (and thus with Russell's Paradox) and draw interesting, non-trivial conclusions. Proof by contradiction is only going to work this way if you're already assuming that contradictions are off the table, meaning you're already rejecting Russell's Paradox before you even do the proof.

Where is yours?

The town barber, who is a man, shaves exactly every man in the town who does not shave himself.

Does the barber shave himself? — Jeremiah

The barber either has a beard, or leaves town in order to get shaven wether or not he does it himself or visits a collegue, for sure he doesn't shave himself when in town.

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.

You do realize this has been a recognized paradox since 1901? — Jeremiah

Argument from authority? really?

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

You do realize argument from authority is only wrong if the authority is wrong. I think over 100 years of history is a very strong authority. — Jeremiah

Nonsense, argument from authority is wrong as long as one of the participants doesn't accept the authority of claimed authority by other(s). You ought to provide the actual argument, not appeal to the authority, especially if the authority gets questioned.

Having said that, personally I'm not fighting your paradox other than in my other post, wich if you read correctly substanciates the perceived paradox, but attempts to force you to improve on its formulation, not questions said authority.