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
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
Have I got this about right? A couple of comments. — fishfry
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
You have not given any argument as to why it must be denied — Jeremiah
The paradox is whether he belongs to the set of men who get shaved. — Jeremiah
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
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
... the supposition there's a largest prime, lacks any persuasive force for it. It doesn't follow from seemingly reasonable principles. — MindForged
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
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
The assumption that there's a largest prime doesn't seem to rest on comparably reasonable principles — MindForged
such as a set being any collection defined by whatever condition you have in mind. — MindForged
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
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
That's what I'm arguing, sort of. Sure, Russell's paradox is a paradox. That was never the dispute. — MindForged
The issue was always that the principles that gave rise to the paradox in naive set theory seem pretty damn reasonable. — MindForged
So the way out of it was to come up with ad hoc restrictions on what constituted a set. — MindForged
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
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
One can get around it the way ZFC does, — MindForged
but the question is if that is more rational or if it results in a more theoretically virtuous theory. — MindForged
At the very least, the Incompleteness Theorems leave the door open about which one you pick, — 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. — fishfry
This is recency bias, not a reasonable explanation IMO. — fishfry
(Answering these together)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?"
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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.
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What do you mean by dismissing the possibility of paradoxes? — fishfry
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
He got the point right away. You agree that Frege himself got the point right away. Yes? — fishfry
It would not be reasonable for a physicist to reject a method that works in practice simply because it lacks mathematical rigor.
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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
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
Certainly different than claiming he cannot exist at all and if he could carry it out then it would not be a paradox. — Jeremiah
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
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