Braininvat » February 24th, 2018, 5:46 pm wrote:Agree. I was wondering if there might be a parallel relation to what this thread is working on, so I tossed this in. Admittedly, a speculative and not well informed toss.
Asparagus » February 19th, 2018, 11:23 pm wrote:The issue is that set theory leads to a contradiction. R is one example of it. I think Russell was saying that the contradiction is the result of the mistake of treating a collection of classes as one class.
But isn't that pretty close to what set theory is doing with natural numbers? I think Russell was questioning the basic assumptions of set theory, not saying that set theory is fine except R can't exist. But I'm totally open to being corrected about that.
Asparagus » February 19th, 2018, 11:23 pm wrote:But consider the attitude that R can't exist because it's contradictory. Would it follow from that that logic reliably guides us in understanding the world?
hyksos » February 25th, 2018, 7:38 pm wrote:Are you contending that R is definitely a set?
hyksos wrote:Are you claiming R is forbidden from mathematics due solely to convention and convenience?
Lomax wrote:Actually the set of {all sets which do not contain themselves} does meet this condition, by virtue of being both.
Lomax » February 26th, 2018, 5:43 am wrote:Hyksos, putting aside that sets are usually undefined in set theory (that is to say, they are "primitives"), I have a problem with your "defining" characteristic, "it can be determined if an element, e, is either in the set S, or not in S". Actually the set of {all sets which do not contain themselves} does meet this condition, by virtue of being both. Do you rather mean to say that "an element, e, cannot both be in the S and not in the set S"? Because if so, that's what's being asked. Set theory doesn't "solve" this problem by just insisting on an answer. It just avoids it by ad hoc means. Which is the most common criticism of Russell's own proposed "solutions".

RJG » February 26th, 2018, 4:29 am wrote:Lomax wrote:Actually the set of {all sets which do not contain themselves} does meet this condition, by virtue of being both.
This is as nonsensical as positing the set of {all sets}.
Since it is logically impossible for sets to contain themselves, then {all sets which do not contain themselves} = {all sets}.
Lomax wrote:RJG, it is not enough to just say it. We are talking about methods of proof here. The point is that saying "it must satisfy the condition of being one or the other" does nothing to advance the proof, particularly bearing in mind that disjunctions can be inclusive.
Hyksos wrote:We have machinery today that Russell did not have during his lifetime. During his lifetime, his solutions may have been completely adhoc, sure.
We might also render it as
(i) ~i
As to whether it can be rendered in predicate logic, I think that would be tougher. We might say we can render it simply as
(i) ExFx
Where the predicate F means "asserts ~ExFx", but perhaps some would argue we've split the statement into two, and changed the nature of the game.
Premise1: ~Premise1
Premise one: ExFx
Asparagus » February 26th, 2018, 6:49 pm wrote:IOW, if you want to make a strong claim that there is no such thing as R, you're going to have to bind yourself to some interesting ontological commitments about math that will lead us straight back to the OP.Hyksos wrote:We have machinery today that Russell did not have during his lifetime. During his lifetime, his solutions may have been completely adhoc, sure.
The paradoxes of naive set theory can be explained in terms of the inconsistent assumption that "all classes are sets". With a rigorous foundation, these paradoxes instead suggest proofs that certain classes are proper (i.e., that they are not sets). For example, Russell's paradox suggests a proof that the class of all sets which do not contain themselves is proper, and the BuraliForti paradox suggests that the class of all ordinal numbers is proper. The paradoxes do not arise with classes because there is no notion of classes containing classes. Otherwise, one could, for example, define a class of all classes that do not contain themselves, which would lead to a Russell paradox for classes. A conglomerate, on the other hand, can have proper classes as members.
hyksos » March 9th, 2018, 1:45 pm wrote:I did not claim "R does not exist." I only demonstrated it was not a set. It could still exist, as some other entity, such as a class.
hyksos » March 9th, 2018, 1:45 pm wrote:!Edit : My apologies up front. On a 2nd reading, it seems like I have not entirely fleshed out my position on this. According to the article linked, the Russell Paradox is strong enough to chase its way up the ladder to classes too. SO that there could be a "Russell Paradox of classes". Hmm...
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