Years ago, I already offered this solution here on ILP, but I’m posting it again because just now, when I was reminded of it, I had a thought that I consider its seal.
Russell’s paradox is about sets that do not contain themselves. I, however, contend that sets by definition contain themselves. A set consisting of all my Lego pieces does not contain the box I keep them in (in fact, I have no Lego pieces, let alone a box to keep them in). A set consisting of all my Lego pieces consists entirely of my Lego pieces. So the set of all my Lego pieces contains itself, as it’s nothing more than all my Lego pieces.
The seal on this solution is this. Let us, in light of the above, consider the set of all sets that do not contain themselves. That set is then an empty set. An empty set does not contain itself, as it’s by definition a set that contains nothing. In a different sense, however, it does contain itself, for the same reason that the set of all my Lego pieces contains itself.
Of course you are correct… if it doesn’t contain itself, it’s not a set, by definition, so the moment someone says that there’s something outside a set that is the set, and that this creates a paradox, they are contradicting themselves.
Even is as a worst case scenario, it was an actual paradox that we couldn’t solve, it would belong to the set of paradoxes and be a member of itself, distinct from the set of non-paradoxes.
Wow… i thought that was easy to understand. Even is a paradox emerges, it belongs to it’s own set, the set of paradoxes… so your argument is still true.
A set is not the contents of itself, it is an abstraction. The set of all the pens on my desk (one) is not a pen on my desk, it’s a logical descriptor that remains constant however many pens are on my desk. The fact that an abstraction isn’t an object… doesn’t make it an object.
But maybe we are using different uses of “is”, or something. So If we have the set of “all things except those beginning with an S in English”, it (a Set) contains itself? That’s still not a paradox? Surely that’s the set of “all things except those beginning with an S in English but including itself as a further exception”, which is a different set entirely.
As I understand it, any definable collection is a set. If Sauwelios collected Legos then he would have a set of Legos; however the set of Legos is not itself a piece in the collection. The plural is not a single. This is a normal set. An abnormal set would be a set of everything that is not a Lego, be it the table, the rooms, the chair, and also the set itself.
Now the paradox comes from trying to figure if a set of all normal sets is normal or abnormal. Based on the definitions established you cannot tell. Since a normal set excludes itself, it cannot be part of this set. An abnormal set can be itself, because it is made of everything else outside of the collection, but it could not normal sets.
For those that think that this is just a matter of definitions, I respectfully disagree. I think that the paradox goes back to the Liar’s paradox. “Everything that I tell you is a lie.” Like the Russell’s paradox, it originates in self-reference. A set of all normal sets, cannot be itself but also cannot be other than itself. Same with the liar’s statement. Everything said is a lie, but to believe this we have to assume that it is true, thus destroying the collection.
For me these paradoxes speak to the limits of knowledge. The eye can see everything, except itself.
That paradox is due to an in my view absurd notion of what it means for a set to contain itself. For instance, take the set of all even prime numbers and itself. You seem to think that its notation would look like this:
{2,{2,{2,[…]}}}
I, however, contend that it would look like this:
{2}
All abstractions are ultimately abstractions from concrete things. Last night, I basically posted my OP on Philosophyforums.com. Within several hours, a mod moved the thread to the Unmoderated forum–which falls under the Not Quite Philosophy section–, saying:
[size=95]“Russells’s Paradox is a mathematical question and will neither be explained nor resolved by piles of words or analogies with Lego.”[/size]
Though he does not quite say it, this suggest to me that he thinks mathematical questions can never be explained or resolved by words or analogies. But without analogies, mathematics remains a mere abstraction, with no connection to existence. And as for words, you may want to compare this post of mine, which I also made last night:
In the article section linked to there, it even says:
[size=95]“English prose is a poor tool for expressing fine logical distinctions (just as it is an unsuitable tool for expressing fine mathematical distinctions[3]).”[/size]
These two things in combination got me thinking. What kind of person favours mathematical or logical notation over natural languages? What kind of person tends to consider “Analytic Philosophy” the only real kind of philosophy (when perhaps it should not even be considered philosophy at all)? What kind of person was Bertrand Russell? Think back to when you were in high school, perhaps even in elementary school. Don’t you remember the guys–they are mostly guys–who were good at mathematics and the like, but bad at languages? The more “exact” it was, the better they were at it: first mathematics, then physics, then chemistry, then probably geology and then biology, etc. Is it not likely that it’s not so much the case that the natural languages are poor tools for expressing fine logical or mathematical distinctions, as that they are poor at using those languages for expressing those distinctions?
I was reminded of Russell’s Paradox when I looked into dialetheism a bit. Arguments for dialetheism especially appeal to Russell’s Paradox and the Liar’s Paradox. I however am impressed by neither. The Liar’s Paradox is simply an implicit contradiction: “[It is true that] this sentence is false.” I agree with what you say about the eye, though, and would add that any kind of self-reference is logically impossible. The hand can never touch itself; at most it’s a part of the hand that touches another part.
You may contend that, but you’d be wrong. You can make your own definitions of sets, of course, and you’re welcome to, but you’ll be talking about different sets to the sets that mathematicians and logicians talk about.
On the other hand, to be fair in applying that line of thinking, the example you gave is not a set according to modern mathematics, as it’s axiomatic that a set cannot contain itself. Making it axiomatic is a useful way of “solving” Russell’s paradox, too. S={2, S}
That doesn’t make them concrete things. They’re still abstractions.
It is in many ways a pure abstraction; possibly the purest; but it is also striking how often the profoundly abstract flights of fancy that mathematics takes are later found to have useful applications in the real world - without any interceding analogies.
At its simplest, it’s a structure of axioms (usually analogous to real life) and a series of rules governing relations. And proofs, solutions, breakthroughs, come from rearranging and reapplying those rules in different permutations. But if you want to critique it, you have to know the rules.
I don’t think that’s necessarily the case; Bertrand Russell is a perfect example of an Analytical philosopher who could write beautifully - witty, careful, elegant prose (whether you agree with the content or not). There are several good expositors of mathematics and the exact sciences (and many bad ones). The fact that most of the best mathematicians in my year at school were not interested in the humanities doesn’t negate the fact that probably the best, who went on to study mathematics at Oxford, was also a gifted organist and is now a published poet.
If you want offensive extremes, continental philosophy appeals to those who value creative acts without any concern for the quality of what sprouts forth, while analyticals prefer not to create anything as they’ll only pick holes in it before the day is out.
“This sentence is six words long” doesn’t work for you?
At most it’s a part of the hand that touches anything, and things only ever touch a part of the hand. That’s what we mean by a hand touching something.
And this is why it is only a paradox of humans perception and reason. Not a paradox about reality.
If the same thing can occupy any number of sets then it is is only a problem of category and not a problem of the thing in itself.
Were it not for those pesky humans trying to put everything in a convenient conceptual box, everything would be fine.
About that implicit contradiction, it is no accident that I chose the version of the paradox where there is a speaker. Adding the implicit to a speaker does not remove the paradox of the self-reference: “It is true that everything I tell you is a lie”. The bind comes in the universal set that develops when the phrase used is “everything” or “all”. The statement contains it’s own negation because it becomes an exception to the everything.
Ultimately though it might be nothing more than a quirk of language, but it raises interesting philosophical questions about just what is “caught” by words like “nothing” or “everything”. Paradoxes are like dead ends on the road-- but just like the roads, it does not mean that all roads are dead end or that the purpose of a road is to lead to nowhere.
You mean it’s (figuratively) axiomatic that a set cannot be truly notated as containing itself, because of the infinite regress? After all, S={2, S} is just another way of writing S={2, {2, {2, […]}}}.
But abstractions ultimately refer to concrete things. There could be no abstractions without (the notion of) concrete things.
Fair enough.
[size=95]“‘[M]ost beautiful’ is not the same as ‘most profound’ and even as ‘most perfect in regard to language’.” (Leo Strauss, "Note on the Plan of Nietzsche’s Beyond Good and Evil.)[/size]
Interesting about the poet. Perhaps he felt that even the most prosaic language–mathematics–was ultimately inadequate?
No, because the phrase “this sentence” does not have a referent when it is spoken (or written, typed, etc.), as it is not a sentence.
Then we can say everything does everything to itself, and leave it at that.
But don’t sets have properties that their contituents might not be allowed to have? Like, the set of all rocks that weigh between 5 and 6 pounds. It seems to me you have two problems- I would think a set wouldn’t weigh anything, and then it doesn’t meet the criteria to be a member of itself (leaving aside that it isn’t a rock, either). Another way you could interpret it though, is that since this set includes a whole lot of rocks, the set itself must weigh a heck of a lot more than 6 pounds and thus can’t be a member of itself for that reason. I’m not sure which of these you agree with, but it seems you must agree with one- I don’t see how this set can weigh between 5 and 6 pounds and thus be a member of itself.
Supposing you have a reason why a set can ignore it's own criteria for membership, what's your reason that other sets can't? If the set of all rocks that weigh between 5 an 6 pounds gets to ignore the criteria and be a member of itself, why can't the set of all hippopautumuses named Steve ignore it too, and be a member of the set of all rocks that weigh between 5 and 6 pounds? I see no reason why one set is more qualified than the other- it certainly wouldn't be the first time a set contained multiple sets, so that can't be an issue.
EDIT: One more problem: What about sets specifically designed to not include themselves? "The set of all sets Uccisore hasn't written about today" pretty clearly doesn''t include itself (clear to me, at any rate). In fact, I mention it for no reason other than to descibe a set that doesn't include itself. I really don't feel like I failed or see any reason why I should have, so where did I go wrong? It gets so bad (the set of all things that aren't sets, etc.) that it seems you're either going to be introducing more paradoxes than solving, or having to just by fiat declare "That isn't really a set" when it's plain to everybody else that it is.
First Russell’s Paradox is merely an oxymoron, a “square-circle”, a self-contradicting definition. Such is only a paradox if you first presume that such can exist. The set that contains all sets that do not contain themselves can never exist in the first place. It is a “circle with four corners”. It is like saying that you have a number that is smaller than half its size or the famous statement, “This statement if false”. It is merely confused rhetoric (a Russell specialty).
I don’t see how you can say that by definition all sets contain themselves. A set of all sets with less than 3 members is a very, very large set and thus cannot include itself. It would be a “normal set”. Conversely the set of all sets that have more than 10 members would certainly include itself. And that would be an “abnormal set”.
And realize that your “set of Lego pieces” is not a Lego piece. The set is not merely a piece like the true members are. So your Lego set is a normal set, excluding itself. It’s members are merely pieces, not sets of pieces (unless you redefine it as such).
Realize that logic “sets” do not have containers. An “empty set” is not an empty box, but rather the same as nothing. Thus an empty set (nothing) does in fact include itself, nothingness.
I think there’s something to both possibilities. If the set of all rocks that weigh between 5 and 6 pounds has more than two members, then it contains at least three subsets of two rocks that weigh between 10 and 12 pounds (the subsets, that is; not the rocks). In fact, this is a problem that James addresses, and that I addressed in my original intimation of my “solution”: http://www.ilovephilosophy.com/viewtopic.php?p=2264654#p2264654. My conclusion in that thread was basically that, if a set of single items cannot contain multiple items, because multiple items are not single items, sets of single items of which there are more than one cannot exist…
As for the other possibility:
This is all sensible, but it does, I think, imply the following. Sets cannot contain concrete things, but at most abstract referents to concrete things. After all, the set of my laptop and the number 1 is nonsense: my laptop is right here, but where is the number 1? The distinction must be made between abstract sets, like in logic or mathematics, and concrete sets.
James, in my view it’s precisely because logic sets do not have containers that they by definition include themselves. After all, if the set {1, 2} has no container, then the set of that set is the set {1, 2}, not {{1, 2}}.
The group of all individual items is not within the group of all individual items because it is not an individual item.
The group of all democrats is not a democrat.
The group of all Lego pieces is not a member of the group of all Legos pieces because the group is not a piece.
Sure, but two democrats are in the group of all democrats. All democrats are in the group of all democrats. The group of all democrats consists of nothing but democrats. Therefore, the group of all democrats contains the group of all democrats. (One democrat is a group of one democrat.)