List of unusual Wikipedia articles(en.wikipedia.org)
en.wikipedia.org
List of unusual Wikipedia articles
http://en.wikipedia.org/wiki/List_of_unusual_articles
13 comments
WTF (A chandelier made of human bones): http://en.wikipedia.org/wiki/File:Sedlec-Ossuary.jpg
I am surprised by the lack of the Brother Metal page, but I'm not sure if it deserves being added or deleted :)
http://en.wikipedia.org/wiki/Cesare_Bonizzi
This reminds me of:
Does the set that contain all sets contain itself?
Does the set that contain all sets contain itself?
Yes, and that's not a contradiction.
The problem comes with the set of all sets that don't contain themselves. Does that set contain itself? Both "yes" and "no" lead to contradiction. That contradiction is the basis of Godel's Incompleteness Theorem, the Halting Problem and others.
The problem comes with the set of all sets that don't contain themselves. Does that set contain itself? Both "yes" and "no" lead to contradiction. That contradiction is the basis of Godel's Incompleteness Theorem, the Halting Problem and others.
The collection of all sets is not a set under ZF set theory, regardless of choice.
Such a collection, if it were a set, would imply the existence of a set of all sets that did not contain themselves from the axiom schema of specification.
Such a collection, if it were a set, would imply the existence of a set of all sets that did not contain themselves from the axiom schema of specification.
If that set is well-defined, it obviously contains itself. Under ZFC, no set can contain itself, so that set is not well-defined.
http://en.wikipedia.org/wiki/Axiom_of_regularity
But there may well be set theories where it is well-defined, hence does contain itself.
http://en.wikipedia.org/wiki/Axiom_of_regularity
But there may well be set theories where it is well-defined, hence does contain itself.
As noted above, the collection of all sets cannot be a set in any set theory with specification, regardless of regularity:
Call the collection of all sets S. We specify T by
T=\{x| x \in S \wedge \neg x \in x\}
T is then the subclass of S of sets that do not contain themselves. Thus with specifiction (provable from replacement, or as an axiom by itself), it is contradictory for S to be a set.
If you reject the law of excluded middle, you can have a intuitionist set theory where S is neither a set nor not a set; alternatively, you can have a set theory without specification one constructed based on a type theory might meet this requirement.
Call the collection of all sets S. We specify T by
T=\{x| x \in S \wedge \neg x \in x\}
T is then the subclass of S of sets that do not contain themselves. Thus with specifiction (provable from replacement, or as an axiom by itself), it is contradictory for S to be a set.
If you reject the law of excluded middle, you can have a intuitionist set theory where S is neither a set nor not a set; alternatively, you can have a set theory without specification one constructed based on a type theory might meet this requirement.
I think lists and articles are distinct entities on Wikipedia, so a list of articles wouldn't contain a list.
Except that they are not. They are called "List Articles", cf. http://en.wikipedia.org/wiki/Wikipedia:Lists#List_articles
Back to the main topic, I think in naive set theory a set of all sets will contains itself. Russell Paradox is problematic when asking the negated statement, i.e. whether a list of all lists that do not contain themselves contains itself.
Back to the main topic, I think in naive set theory a set of all sets will contains itself. Russell Paradox is problematic when asking the negated statement, i.e. whether a list of all lists that do not contain themselves contains itself.
One of my favorite Wikipedia categories: http://en.wikipedia.org/wiki/Category:Lists_of_lists
Bir Tawil would be a nice place for a data haven, were it not landlocked.
Hey, I know, let’s open a data haven on the Moon: http://en.wikipedia.org/wiki/Terra_nullius (Oh, the joys of Wikipedia :)
well there goes the next 4 hours of my life. the math ones are pretty neat - http://en.wikipedia.org/wiki/Vampire_number