If you don't know what a paradox is, then maybe its not a good thing to think about the next paradox. Anyway, a paradox is a set of facts creating a nearly unsolvable puzzle.
Okay, the paradox is as follows and onnly resists whitin the quote.
Quote from: "Paradox"
Fact 1: The next line inside this quote is always false.
Fact 2: The previous line inside this quote is always right.
The paradox here is that this seems to be impossible. Can you figure out how it works?
All paradoxes are supposed to be impossible. A paradox is a contradiction, that is, something that, at the same time, is true and false.
I will rephrase the paradox a little, because I think there's something wrong with it. It should read:
Quote
Fact 1: The next line inside this quote is false.
Fact 2. The previous line inside this quote is true.
I'll explain why I rephrased it afterwards.
The paradox works like this:
Suppose Fact 1 is true. Fact 1 states that "The next line inside this quote is false". Therefore, as Fact 1 is true, the next line, which is Fact 2, is false. Therefore, Fact 2 is false.
Now, Fact 2 states: "The previous line inside this quote is true. But we know now that Fact 2 is false. Therefore, it is not true that Fact 1 (the previous line) is true. If Fact 1 is not true, then it's false. So, we assumed in the beginning that Fact 1 was true, and reached the conclusion that it's false! That's a contradiction!
Ok, maybe we arrived at this contradiction, because the truth is that Fact 1 is false. If Fact 1 is false, it's not true that "The next line inside this quote is false". That means Fact 2 is not false, that is, Fact 2 is true. But if Fact 2 is true, then "The previous line inside this quote is true", therefore Fact 1 is true. However, we had assumed that Fact 1 was false.
So, if Fact 1 is true, then Fact 1 is false.
And if Fact 1 is false, then Fact 1 is true.
That is impossible. Fact 1 can be neither true nor false :D
Why did I rephrase the paradox?
Well, first of all, I prefered to use the term "true" instead of "right", to make it simple and to call things like they are usually called in Logics. Also, I couldn't make the paradox work with the word "always" in it. I mean, the negation of "The previous line inside this quote is always right" is "The previous line inside this quote is not always right". And that doesn't mean that "The previous line inside this quote is always false". Get it?
Other similar paradoxes:
A man says he's lying. Is he telling the truth or not?
Another:
"Yields falsehood when prededed by its quotation" yields falsehood when preceded by its quotation.
There are more at Wikipedia, I think.
I remember having read a long time ago an african tale, about a tribe who let its prisoner decide the question they want. Then, if the prisoner answered right, they sacrifice him to the idol of Truth, and if his answer was wrong, to the idol of Lie.
One day, they catch a very clever boy. That was the question he decide they would make him: "To wich idol I'm going to be sacrificed?". They make the question, and then he answer: "To the idol of Lie".
So, they have to free him and destroy the idols.
Quote from: "rncekel"I remember having read a long time ago an african tale, about a tribe who let its prisoner decide the question they want. Then, if the prisoner answered right, they sacrifice him to the idol of Truth, and if his answer was wrong, to the idol of Lie.
One day, they catch a very clever boy. That was the question he decide they would make him: "To wich idol I'm going to be sacrificed?". They make the question, and then he answer: "To the idol of Lie".
So, they have to free him and destroy the idols.
Nice story :D
I used to tell people that "I'm a compulsive liar", then when they said "really?", I said "No".
Quote from: "KrazyKatz"I used to tell people that "I'm a compulsive liar", then when they said "really?", I said "No".
Had that experience.
PPH, Where did you coppy that from, in other words, what did you smoke man? Don`t tell me you actualy set there and thought about it. That would be... well.. ehm... :)
Quote from: "XAVT"Quote from: "KrazyKatz"I used to tell people that "I'm a compulsive liar", then when they said "really?", I said "No".
Had that experience.
PPH, Where did you coppy that from, in other words, what did you smoke man? Don`t tell me you actualy set there and thought about it. That would be... well.. ehm... :)
Well, I already knew paradoxes like that, so I didn't have to think much. I had already done all the thinking! LOL Well, I did think it so that I could explain it. And yes, you're right. That is a little freaky. But I like that kind of stuff (Logics, Math and all that).
While preparing for an exam on Algebra, I have had to explain to one of my sons the following mathematical paradox:
Let G be the set of sets that don't include themselves.
Then G MUST include itself and MUST NOT include itself
Yeah, that's Russell's paradox. A guy named Frege had developed a whole mathematical theory based on the sets of all sets. I mean, he wrote several volumes about that theory. And Bertrand Russel destroyed his whole theory with just that paradox.
I'm a freak, I know.
EDIT: I misread. Frege's theory was based on the "set of all sets". The explanation of why that set doesn't exist includes a paradox similar to the one you have written about.
EDIT2:
http://en.wikipedia.org/wiki/Russel%27s_paradox
There's a little difference between Russell's paradox and rncneckel's. In Russell's paradox, it's the "set of all sets that don't contain themselves as members (that is, elements), while rncneckel's formulation is about a sets that don't contain themselves as subsets.
Quote from: "PPH"There's a little difference between Russell's paradox and rncneckel's. In Russell's paradox, it's the "set of all sets that don't contain themselves as members (that is, elements), while rncneckel's formulation is about a sets that don't contain themselves as subsets.
I think you misinterpreted rncekel's paradox a bit; isn't every set a subset of itself ;)
Quote from: "Relabsoluness"Quote from: "PPH"There's a little difference between Russell's paradox and rncneckel's. In Russell's paradox, it's the "set of all sets that don't contain themselves as members (that is, elements), while rncneckel's formulation is about a sets that don't contain themselves as subsets.
I think you misinterpreted rncekel's paradox a bit; isn't every set a subset of itself ;)
Yes. But not all set is an element of itself, that is, contain itself as a member. That's the difference between the paradox I'm talking about and rncekel's (yes, in the beginning I misinterpreted it).
Quote from: "PPH"Yes. But not all set is an element of itself, that is, contain itself as a member. That's the difference between the paradox I'm talking about and rncekel's (yes, in the beginning I misinterpreted it).
To me rncekel's paradox still seems exactly the same as Russell's paradox. What's the point in
Quote'Let G be the set of sets that don't include themselves.'
if the 'include' means the set being subset of itself because in that case G is empty set without any paradox, no?
Yes, I agree that his is Russell's paradox or, at least, it was Russell's paradox that he meant. Still, "include" is used (at least in other language) when one set is a subset of another, not when one set is an element of another (at Wikipedia they say "included as a member" to avoid confusion; in Spanish we use the word "belongs" instead of "is included" in the case of elements).
But even when "include" had that meaning, the set G couldn't possibly exist, since the empty set also includes itself (all sets include the empty set, right?).
All sets include themselves. Therefore, the set of sets that don't include themselves is empty. But the empty set also includes itself. Therefore, G cannot exist.
I mean, the empty set is empty because it doesn't have any elements. But it has one subset: itself.
Quote from: "PPH"But even when "include" had that meaning, the set G couldn't possibly exist, since the empty set also includes itself (all sets include the empty set, right?).
All sets include themselves. Therefore, the set of sets that don't include themselves is empty. But the empty set also includes itself. Therefore, G cannot exist.
I still don't find anything wrong with G being empty :). Even empty set includes itself, as you said, but that's why empty set is not _an element_ in set G and G is indeed empty.
(could we get 'mathematical alphabets' to the forum :P )
Quote from: "Relabsoluness"Quote from: "PPH"But even when "include" had that meaning, the set G couldn't possibly exist, since the empty set also includes itself (all sets include the empty set, right?).
All sets include themselves. Therefore, the set of sets that don't include themselves is empty. But the empty set also includes itself. Therefore, G cannot exist.
I still don't find anything wrong with G being empty :). Even empty set includes itself, as you said, but that's why empty set is not _an element_ in set G and G is indeed empty.
(could we get 'mathematical alphabets' to the forum :P )
I just realized I misinterpreted your first post (I just reread my quote in that post, and realized what you were talking about). You're right: I must have misintepreted rncneckel's post. If he were talking about subsets instead of elements, G would be the empty set and there would be no paradox.
(and yeah, my last post was stupid :D I got confused and mixed the "element" and "subset" thing and wrote a preposterous conclusion)
Quote from: "Relabsoluness"(could we get 'mathematical alphabets' to the forum)
If it would be possible to use HTML in a post, mathematical characters can be found in Unicode-set 2200.
Quote from: "Sam_Zen"Quote from: "Relabsoluness"(could we get 'mathematical alphabets' to the forum)
If it would be possible to use HTML in a post, mathematical characters can be found in Unicode-set 2200.
You can also make an image and post it that way ;)