Maybe restricted quantification is not ad hoc at all though. Maybe "Some Gs are F" is a better paradigm than "Some xs are F".

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. — Jeremiah

I can understand your frustration about this, but please don't box me into that group on forehand. The fact that I disagreed with an argument used doesn't nessesarily mean I disagree with it's conclusion. If I disagree with a conclusion, I'll provide an argument leading to a different conclusion.

I can also understand why you do it. However in my opinion this is a debating tactic, wich has no place in discussions. On the other hand, it's quite possible that the persons you usually use this tactic on made a comment that has no place in a discussion either, so I can understand why you choose to do so.
If you can get away with it, it's usually a sign the person you used it to wasn't discussing either, but instead was debating. So I would say that its effective in a debate, but has no place in a discussion. Though of course, when having people in your discussion that don't know how to discuss and are debating instead, it can be an easy tool to get rid of them.
Personally I prefer to stick to the argument, as long as they don't provide actual counter arguments, but come up with a fallacy instead, I merely point out the fallacy rather than trump it with another fallacy. But again, if it works for you, fine, just don't try it on me, I resent debates, I love discussions.

Now lets get back to the barber paradox. The way you formulated it, it seems to have a backdoor.

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 backdoor lies in that it's not clear wether the 'in the town' part refers to that the shaving happens in town, or that it refers to the men living in the town. So to close it, i suggest either a formulation like :

"The town barber, who is a man who never leaves town, shaves exactly every man in the town who does not shave himself."

or

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

Otherwise "the barber shaves himself when out of town" seems to be a valid solution. The second formulation isn't airtight either though, since the town barber could be someone living in another town.

For the record, the barber can't exist, the Russell set we (most of us) don't allow to exist, but the word "heterological" does certainly exist and can go to hell.

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.

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.

It just means this is a task the barber cannot carry out. It is an invalid way of specifying a task.

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

What's curious is that if you consider these two commands

(a) Shave all and only those who do not shave themselves; and

(b) Form a set by selecting as members all and only sets that do not contain themselves as members,

then many people will conclude it is impossible to obey (a), but are confused by (b) and think it should be possible.

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.

Agreed, except that the existence of such a set is a presupposition, and it is that presupposition that must be denied. (In this way it's analogous to being asked if the present king of France is bald, or if you've stopped beating your children.}

https://plato.stanford.edu/entries/mathematics-inconsistent — MindForged

Ahhhh, very interesting article. I learned something.

I do feel a tiny bit sandbagged in the sense that you've had this somewhat obscure topic in mind as you've been debating. Had you presented this article and its point of view up front, it would have made your posts much more clear to me. Minor issue, now I'm educated and I see what you're talking about.

To summarize the article as I understood it:

* We really really really want to save naive set theory, so we have to rehabilitate unrestricted set formation via predicates, aka unrestricted comprehension.

* The reason we care is that we [not me, actually, the people doing this work] would like to rehabilitate logicism, the idea that math is derivable from logic.

* Unrestricted comprehension leads to a contradiction, and in standard logic a contradiction implies any given proposition. That's the principle of explosion. So we need to abandon explosion.

* For various technical reasons we need to also abandon or modify some other logical principles.

* Once we've done this, we can in fact allow unrestricted comprehension and save naive set theory and perhaps even logicism. Although in my opinion you're wrecking logic to save logicism, which might arguably be self-defeating. Nevertheless, this work can be done.

* Now having saved unrestricted comprehension and perhaps logicism [at the expense of wrecking logic IMO] we can also patch up standard math: number theory, analysis, topology, and so forth. Surprisingly, quite a bit of math can be preserved even at the expense of allowing the contradiction of unrestricted comprehension.

* This project is relatively new, and work continues as we speak.

Have I got this about right? A couple of comments.

First, this does remind me a bit of the constructivist project to rebuild math with a countable set of real numbers, each of which can be explicitly constructed. A lot of classical theorems fail in this scenario, so the constructivists patch and hammer and sing and dance and try to fix everything up.

Yes it's true that it's all logically correct, but it seems like so much trouble just to avoid the truths of 20th century math: that unrestricted comprehension fails and that there really are important mathematical objects that can be proven to exist but that can not be explicitly constructed.

So yes, the paraconsistent project is interesting and I'm sure the professors are getting their grants and doing their work and getting tenure and serving on academic committees and having fine old careers.

But if I met one of these distinguished characters, I would ask them the same question I've asked you and @Jeremiah: Why don't you hack logic to allow the existence of a largest prime? Why does one easily proved mathematical fact annoy you so much yet you accept the proof of the infinitude of primes? [Sorry didn't mean to imply you personally are annoyed, you already said you're not. I mean the generic "you," the people trying to rehabilitate naive set theory].

You know we could create a system of math with only finitely many primes. For example let 7 be the largest prime. We want the fundamental theorem of arithmetic (unique factorization into prime powers) to be true. So 1, 2, 3, 4 = 2 x 2, 5, 6 = 2 x 3, 7, 8, 9, and 10 are allowable numbers. But 11 falsifies the FTA, so it's abolished from the number system. 12 ok, 13 is abolished, 14's ok, 15 and 16 are ok, 17 is abolished. And so forth.

Now we have a system of arithmetic that obeys the fundamental theorem of arithmetic and in which there is a largest prime. There's a little problem, which is that the integers are no longer closed under addition, since for example 8 is a number and 9 is a number but 8 + 9 = 17 is no longer a number. Well I guess we'll just drop the rule that says the integers are closed under addition. You're already perfectly willing to abolish the truth table for material implication which says that False implies True, and rejecting the additive closure of the integers doesn't seem much worse.

But notice that we can still preserve the fact that the integers are closed under multiplication! Any product of powers of 2, 3, 5, and 7 is also such a project. See, we are making progress! With a little effort we can probably make this system work very nicely with a few such modifications.

You might object that abolishing 11 will cause practical problems in the world. I agree with that point, and I only used 7 as a simple example. In practice we can just take the largest number anyone could possibly care about, say maybe 10^80, the number of hydrogen atoms in the observable universe, or maybe Graham's number, or Skewes's number. Any old finite number that's so big that nobody could ever care much about it in real life. Then take the next prime after that, define that as the largest prime, and I claim this is a perfectly serviceable system of arithmetic.

So why does everyone care so much about naive set theory but nobody cares about hacking logic and math to allow a largest prime?

I would ask these guys this question if I ever met them.

Now I will allow that I may be one of those old dinosaurs that has to die so that younger people can simply grow up accepting inconsistent math. And I certainly agree that non-Euclidean geometry, relativity and quantum theory, Heisenberg's uncertainty and Gödel's incompleteness, postmodern philosophy and the ills of late-stage capitalism have brought the project of western rationality to a moment of crisis. There's no point defending rationality when the world is so clearly irrational. I take all these points.

But still. Why unrestricted comprehension and not a largest prime? Why the emotional attachment to naive set theory? Maybe my professors were too effective at beating standard mathematics into my brain. But I really don't get it. Naive set theory is intuitively appealing but it fails. Accept it and move on.

Thanks for the link though, I certainly did find it interesting.

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

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.

Have I got this about right? A couple of comments. — fishfry

Well summarized, couldn't have done it better myself. My only quibble is that it's "wrecking logic" only insofar as one already has an idea of what the correct logic is beforehand. The Thomists believed those who started using Classic Logic post-Frege were "wrecking logic" by abandoning what Aristotle left for us, but no one ought take that seriously in light of all the standard mathematics can do for us! Granted, paraconsistent mathematics hasn't reached that level (yet, perhaps) so my comparison probably lacks the persuasive force I'd like it to have. :-) Also, I didn't intend to sandbag you, I should have linked it earlier, though I do believe I referenced paraconsistent math.

Why don't you hack logic to allow the existence of a largest prime? Why does one easily proved mathematical fact annoy you so much yet you accept the proof of the infinitude of primes? — fishfry

Because a paradox is not simply a contradiction. The contradiction "It's raining outside and it's not raining outside", as with the supposition there's a largest prime, lacks any persuasive force for it. It doesn't follow from seemingly reasonable principles. Frege believed unrestricted comprehension was "self-evident", though I loathe that term. But of course we know that the very logic he created cannot be paired with that principle of set theory. So we have two choices on offer: remove/rework the principle as best we can or switch logics. At the very least, the Incompleteness Theorems leave the door open about which one you pick, as in either case you will necessarily lose a very desirable trait for a mathematical system (either completeness or consistency).

Dialetheists hold that classical math/logic fails to account for some pretty crucial data: the truth-value of the Liar sentences, the geometry of Escher spaces, etc. They trivialize one you try to consider such things. Or take unsolved problems that are unprovable (or at least unproven) and we know standard maths leaves one wanting where answers are concerned (Continuum Hypothesis, for example). However, they argue that paraconsistent logics can handle these elegantly and give some real answers, but you can't do it while making an appeal to retain the standard formalism. And so, keeping naive set theory but changing the logic lets you retain a principle that seems very reasonable and you have an explanation for certain data and a reason to give about why some principles and inference rules in standard maths ought to be dropped.

It's a philosophy of science issue, I suppose. Which theory is more theoretically virtuous? That's what they hang their hats on, perhaps.

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

Let's talk about the Barber.

Suppose we have a town with three men: a barber (B), a philosopher (P) who doesn't shave himself, and a mathematician (M) who does.

Now define a set R as all and only men who shave all and only men who don't shave themselves.

1. M is never a member of R because he shaves a man who shaves himself.
2. P can't be a member either because he doesn't shave himself, so he'd have to shave himself to be a member, but he doesn't.
3. What about B? He would have to shave P and not M. No problem. If he shaves himself, he'd be out, like M, but if he doesn't, he'd be out like P. So B can't be a member no matter what he does.

So R = { }. No one shaves all and only men who do not shave themselves, therefore the barber does not shave all and only men who do not shave themselves. The three cases are exhaustive, in fact: no one can be a member of R whether they shave themselves or not.

Your presentation is to start by defining R as {B}, and then saying

The paradox is whether he belongs to the set of men who get shaved. — Jeremiah

But we've already seen that B cannot be a member of R, so the premise is just false.

Now what about Russell? In the analysis given above, R was not the Russell set, but the set of all Russell sets, and it has been shown to be empty. It does not contain any set that contains all and only sets that do not contain themselves, because there can be no such set.

Therefore if you present the paradox by beginning, "Let S be the set of all sets that do not contain themselves as members," then I will deny the premise. No set can be formed in this way, which is exactly Russell's point.

My only quibble is that it's "wrecking logic" only insofar as one already has an idea of what the correct logic is beforehand. The Thomists believed those who started using Classic Logic post-Frege were "wrecking logic" by abandoning what Aristotle left for us ..! — MindForged

I plead tragic ignorance of Aristotelian logic. Perhaps I over-identify the word logic with the standard predicate logic used in mathematics. The paraconsitentists (that word is used in the SEP article I believe) are wrecking what I think of as logic, but clearly my perspective is too narrow.

Granted, paraconsistent mathematics hasn't reached that level (yet, perhaps) so my comparison probably lacks the persuasive force I'd like it to have. — MindForged

Appreciate that! Of course that doesn't mean that 20 or 30 years from now we won't be teaching paraconsistent logic to the undergrads. But it doesn't have much debating force today. You can't sensibly say, "Ok, our assumption of X has led us to a contradiction, so X might be true if we abolish the principle of explosion and tweak a few other things in logic." We don't say that. We say, "We have just shown that X is false."

Of course specialists in logic-tweaking may bend the rules to allow that X is true. But it's hard to argue that this is how we should think. If X leads to a contradiction, X gets rejected.

... the supposition there's a largest prime, lacks any persuasive force for it. It doesn't follow from seemingly reasonable principles. — MindForged

Oh but it does. The primes get exceedingly rare on average the farther out you go. And there are arbitrarily long runs of primes. You name a number n, and I'll show you a run of n consecutive composite numbers. I find it perfectly reasonable that back in Euclid's time, nobody knew whether there was a largest prime, and many learned and brilliant thinkers might have believed that there is a largest one.

And I have seen for myself that many students still ask this question. The infinitude of primes is NOT obvious at all. Of course once one has seen the proof and has fully internalized the infinitude of primes, one can no longer conceived of anyone else's doubt on the matter. But before one proves otherwise, it's perfectly sensible that there might be a largest prime.

Ok next post is the response I've been putting off so let me just post that so at least I can feel like I've caught up.

Ok herewith my response to the deferred post.

Why is it that in the case of (a) you regard this as a basic mathematical truth; yet in the case of (b) you regard this as a philosophical conundrum perhaps susceptible to attack via paraconsistent logic?
— fishfry

Because in the case of A, we have every reason to believe we are in a consistent domain (that of classical mathematics), where proof by contradiction is necessary (on pain of triviality), and we know we can give examples of larger primes . In B, we get a paradox unless we rewrite the rules of naive set theory to get something like ZFC. — MindForged

But no, you're just restating your bias, not explaining it. "Q: Why is primes a simple proof by contradiction, and sets a paradox? A: Because primes is a simple proof by contradiction, and sets is a paradox." You have not explained your position, you've only rephrased it. We "rewrite the rules of set theory?" Well we "rewrite the rules of primes" to outlaw a largest prime, once we see there isn't one.

This is recency bias, not a reasonable explanation IMO.

With A, we have a counter example that let's us dismiss the initial supposition, with B we get a contradiction from what seem like reasonable assumptions on their face. — MindForged

Unrestricted comprehension "seems reasonable" till we prove it's not. You're privileging an incorrect intuition and saying, "Who are you going to believe, an absolute logical proof, or my vague intuitions?"

Before Euclid there may well have been a strong intuition that there is a largest prime. There ARE in fact good heuristic reasons for believing so, which I've mentioned a couple of times.

The assumption that there's a largest prime doesn't seem to rest on comparably reasonable principles — MindForged

But it does. The farther out we go in the integers, the more rare primes become. And there are arbitrarily large runs of composites. Before one receives any mathematical training, it's perfectly reasonable that there are only finitely many primes; and in fact this question does come up among the mathematically naive.

such as a set being any collection defined by whatever condition you have in mind. — MindForged

Nice intuition, turns out to be false. No reason to privilege this intuitive error. You assume it and you derive a contradiction, so it's false.

I assume (although you have not confirmed this) that you don't regard the infinitude of primes as being subject to modification or revision based on paraconsistent logic. Why is (b) different?
— fishfry

I don't think the infinitude of primes will be much affected by a transition in the logic. Paraconsistent logic dispenses with proof by contradiction and tends to instead rely on proof by non-triviality (these are identical in other logics but not with PLs). — MindForged

You can make this argument when paraconsistent logic gains mindshare. I already showed that by dispensing with the principle of explosion and making a few minor tweaks to number theory, we can let 7 be the largest prime and things work out fine. Just as you can crowbar naive comprehension into submission if you're willing to tweak the rules of logic.

Assume the contrary, derive a contradiction, learn a truth.
— fishfry

I suppose the simplest way is to point out there are other concerns that bear on something besides consistency. I can't remember if it was in this thread that I mentioned this, but for example it's just a fact that the early calculus was inconsistent. One had to treat infinitesimals as a non-zero value at one step of proofs and then treat them as having and value of zero at another step of the same proof. This was acknowledged by Newton, Leibniz, criticized by Berkeley, etc., and it remained that way for more than 150 years. Now as far as I can tell, if you really tried to insist on this way of proceeding, you would have been rationally required by your standards to have rejected calculus (and therefore everything learned and built because of it) during that century and a half of it being inconsistent. But that's obviously ridiculous, there are other theoretical virtues besides consistency which made calculus tenable to accept despite the contradictions it required one to adopt. — MindForged

Of everything you've written, this is the one point that made me stop and think. It's a good point. I have a response.

Newton was doing physics, not math. He had a method that worked to give him correct answers, but as Berkeley pointed out, Newton did not have a rigorous mathematical justification for his method of fluxions. We have a modern parallel in renormalization, for which Feynmann, Tomonaga and Schwinger got the Nobel prize. At the time, they had no mathematical justification. I believe the mathematical rigorization of renormalization is a relatively recent development.

The moral of the story is simply that physics leads mathematics by decades or even centuries. Physicists leap in where mathematicians fear to tread.

It would not be reasonable for a physicist to reject a method that works in practice simply because it lacks mathematical rigor. "Lacking mathematical rigor" describes a lot of physics even today. Physicists think in infinitesimals, yet the theory of limits rejects infinitesimals.

And for what it's worth, nobody rejected calculus; but they worked very hard for 200 years after Newton to get it straightened out. Even so, the theory of limits is a bit of a kludge. It depends crucially on the completeness of the real numbers, something for which there is no known analog in the physical world. It's fair to say that the underlying philosophical problem is still open.

This is a mystery, not a paradox. Those are different things.

That's what I'm arguing, sort of. Sure, Russell's paradox is a paradox. That was never the dispute. — MindForged

Arggg! That's EXACTLY what I'm disputing. And even though it's still called a paradox, nobody treats it that way. We treat it as a rigorous and convincing demonstration that naive comprehension must be rejected.

The issue was always that the principles that gave rise to the paradox in naive set theory seem pretty damn reasonable. — MindForged

Falling back on naive intuition again. The Banach-Tarski paradox seems unreasonable, but it's mathematically true and is nothing more than a clever repackaging of the fact that the group of rigid motions of three space contains a copy of the free group on two letters. The proof sketch given in Wikipedia is actually quite simple. Nobody doubts its truth. We just note that "math isn't physics" and move on. By the way this is yet another perfectly correct theorem that's NAMED a paradox that actually ISN'T a paradox. It's simply an intuition-defying demonstration. Math is full of them.

The entire history of math is the triumph of rigorous demonstration over naive intuition. It's only when it comes to set formation that some (you and @Jeremiah and maybe a few others) dig in your heels and say, "No, my naive intuition is more true than mathematical proof." The mathematical commmunity does not share that view. In the course of studying math, many naive intutions are shattered and replace by proof. Naive set theory is just one of them.

So the way out of it was to come up with ad hoc restrictions on what constituted a set. — MindForged

Not ad hoc at all, but rather the product of over thirty years, say from 1900 to the 1930's, give or take, that the modern axioms of ZF were developed. The process was anything BUT ad hoc, and again I would refer you to Maddy, Believing the Axioms.

There were extra-mathematical considerations which led to that response, not simply a proof by contradiction because that argument itself relies on already dismissing the possibility of paradoxes, which is the very thing under dispute of you accept Russell's Paradox. — MindForged

Where would I start here? You've already said you do not reject the law of the excluded middle. So there are not extra-mathematical considerations. You assume a proposition and show it leads to a contradiction, hence the proposition is false, no matter how intuitively appealing it seemed five minutes ago. Poor Frege. He got the point right away. You agree that Frege himself got the point right away. Yes?

What do you mean by dismissing the possibility of paradoxes? Is Euclid's proof of the infinitute of primes a paradox? No, it's simply a demonstration that a common belief (that the primes are finite in number) is false. You say nobody believes this, but I spend a lot of time on Quora and Reddit and this question DOES come up often among beginners.

There has to be a reason (besides arguing against the conclusion) for why you reject the principles that give rise to the paradox, otherwise it seems like the objection is circular. — MindForged

I could not understand that remark. What principles? The pattern is clear. If an assumption leads to a contradiction, we must reject the assumption, no matter how intuitively appealing.

One can get around it the way ZFC does, — MindForged

One "gets around" the finitude of primes by accepting their infinitude. You are simply using different words to describe two identical phenomena. Two proofs by contradiction.

but the question is if that is more rational or if it results in a more theoretically virtuous theory. — MindForged

It is never a question of virtue, but only of truth and proof. [Two different things in general, but in this instance, the same]. There are infinitely many primes and naive comprehension fails.

At the very least, the Incompleteness Theorems leave the door open about which one you pick, — MindForged

Ah ... a while back you objected that I misquoted you saying that incompleteness was on point here. But in fact I believe I was originally correct. You think this is about incompleteness. It's not. In incompleteness we fix a given system of logic (first-order predicate logic in fact) and draw conclusions about sets of axioms. In paraconsistent logic we alter the logical rules to obtain different theorems. That is not the same thing at all.

Appreciate that! Of course that doesn't mean that 20 or 30 years from now we won't be teaching paraconsistent logic to the undergrads. But it doesn't have much debating force today. — fishfry

In that bit, I was referring to Paraconsistent logic lacking as many practical applications (in comparison to standard maths) that would serve to justify it's use. It still has such practical uses (models of human reasoning, databases and some digital logic stuff) and uses in mathematics where standard maths drops the ball. It's just not as inherently persuasive given how much more standard maths does across a broader spectrum of applications.

This is recency bias, not a reasonable explanation IMO. — fishfry

Not really. That primes might be finite would take a lot of work to justify believing, accepting that sets are extensions of properties seems far simpler.

Unrestricted comprehension "seems reasonable" till we prove it's not. You're privileging an incorrect intuition and saying, "Who are you going to believe, an absolute logical proof, or my vague intuitions?"
...
Nice intuition, turns out to be false. No reason to privilege this intuitive error. You assume it and you derive a contradiction, so it's false.
...
What do you mean by dismissing the possibility of paradoxes? — fishfry

(Answering these together)

This seems like the circular reasoning I mentioned before (and this will answer a similar questions you ask elsewhere in your post). You are in effect pre-excluding the the axiom when the question is exactly if the axiom is acceptable. If I'm entertaining the possibility of accepting Russell's Paradox then pointing out the contradiction obviously isn't the defeater for me. Yes, it leads to a contradiction but to avoid it you have to spell out *why* the principles causing the contradiction are to be rejected. We know there are more primes beyond whatever prime one thinks is the largest one. They become rarer, not nonexistent. That's evidence against the view that there's a largest one. So in attempting to reject unrestricted comprehension, you have to have something beyond pointing out the obvious fact that there's a contradiction. We want to know why something has gone wrong (if indeed it has).
And on that point:

Arggg! That's EXACTLY what I'm disputing. And even though it's still called a paradox, nobody treats it that way. We treat it as a rigorous and convincing demonstration that naive comprehension must be rejected. — fishfry

This seems a little silly. Russell's Paradox is a paradox *in naive set theory*. When I said no one disputes that, I meant simply that it was the conclusion of the rules of naive set theory, but it wasn't clear why those rules were faulty beyond "we get a contradiction". We knew there was a contradiction, what made (and makes) the Russell Set a paradox in naive set theory is that the principles that give rise to it seem reasonable, a standard mathematicians employ routinely. We can avoid it of course, but in doing so Zermelo was explicit that he was trying to avoid the paradox in a new theory, not solve it (i.e. provide an explanation for why the axiom separation is a more reasonable axiom). This plays off my earlier comparison, which I'd like you to address:

I can avoid the Liar Paradox. It's easy. I'll simply define a formal language (because natural languages are known to be inconsistent) and by some means disallow self-reference. Poof, no more Liar paradox. After all, via a simple proof by contradiction I know something has gone wrong and so some assumption must be discharged. So I dismissed a feature necessary to create the paradox and voila.


This is obviously missing the point, in exactly the way you did. Evading it is easy, giving a justification for how you did so and why your methid ought to br adopted is not easy. That's what a paradox is, that's why tossing up a proof by contradiction is silly. There are consequences to whatever method you use to dissolve such issues and dismissing self-reference has a large number of negative repercussions, far more than a contradiction does. Sometimes those black marks are acceptable, but that has to be shown and a proof by contradiction does not do that. That's my point in referencing theoretical virtues. The theory resulting from the change in axioms has to be assessed for its worth and consequences. If you address nothing else, this is the main point I'd like to see you tackle.

He got the point right away. You agree that Frege himself got the point right away. Yes? — fishfry

Sure? Think I already said so. But as I also said, Frege was reasoning with the classical logic he'd just created and had no other logical theory (besides the old Syllogistic) with which to explore alternate possibilities. Classically, Russell's Paradox is unacceptable because the set theory trivializes everything, but we know that's possible to contain with Paraconsistency, even if you don't go to dialetheism.

It would not be reasonable for a physicist to reject a method that works in practice simply because it lacks mathematical rigor.
...
And for what it's worth, nobody rejected calculus; but they worked very hard for 200 years after Newton to get it straightened out. — fishfry

This is exactly the point I was making. Calculus was so useful and good a theory that in spite of known contradictions, it was more reasonable to keep it. But you didn't really answer the problem I posed. Neither of us disputes that calculus was inconsistent for a while. On your view, it seems like you'd have no recourse by to dismiss the theory on grounds of inconsistency. Pointing out that physics blazes the trail is besides the point; physicists still use logic. In effect, I can't see how your view isn't inherently flawed because a single contradiction justifies any change needed to avoid it. You certainly didn't reference any limit on what one is rationally committed to doing when faced with a contradiction. So without further explanation, you'd be committed to dispensing with it and other developments depending on it. Ditto for any inconsistent scientific theory.

Ah ... a while back you objected that I misquoted you saying that incompleteness was on point here. But in fact I believe I was originally correct. You think this is about incompleteness. It's not. In incompleteness we fix a given system of logic (first-order predicate logic in fact) and draw conclusions about sets of axioms. In paraconsistent logic we alter the logical rules to obtain different theorems. That is not the same thing at all. — fishfry

I think you are mixing up points. Earlier I referenced how Russell's Paradox led to a reformulation of set theory and I further said that in spite of this reformulation, the Incompleteness Theorems still left the door open to Paraconsistent logic. After all, paraconsistent logics can tolerate inconsistency but, as per Gödel's theorems, the resulting system will be complete (the opposite of standard maths).

An empty set is still a set.

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.

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

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

He's saying that there cannot exist a barber who shaves all and only men who do not shave themselves. No barber is a "Russell barber".

And so there cannot exist a set that contains all sets that are not members of themselves. No set is a "Russell set".

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 — Jeremiah

Let S be a square triangle. How many sides does S have?

Is this a paradox? If not, what makes Russell's set different? I say nothing. In both cases there's just a contradictory premise, and so the "paradox" is resolved by dismissing that premise as incoherent.

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

S being a square circle isn't a paradox. It's just nonsense.
R being a set that contains all sets that do not contain themselves isn't a paradox. It's just nonsense.

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 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.

This means either our math is fundamentally wrong or it is incomplete. — Jeremiah

It just means that the axiom schema of unrestricted comprehension – $∃y∀x(x \in y ⇔ P(x))$ – is inconsistent, and so additional qualifications are required to maintain consistency.