Russell's Paradox

[mendhak]

And I quote


Russell's paradox is the most famous of the logical or set-theoretical paradoxes. The paradox arises within naive set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox.

While I have understood this, I was wondering of some examples of "sets of sets which are not members of themselves."

Off the top of my head, {NULL} would be an example. Is this correct? (How do I type that null symbol, where's NoteMe when you need him? :D )

Can I see more examples of such sets?

[wossname]

Those pesky naive set theories eh?

How about {infinity}?

[Acidic]

If I am to make a catalog of books in a library I don't include the catalog itself do I, I wound't because the reader would know where it is as they are currently reading it.

If I was now to make a catalog of all catalogs not listing themselves I would reach a paradox. If I don't include my own catalog, it is not complete. If I do include my own catalog, it does include itself, thereby making the catalog invalid.

Does that qualify?

[kedaman]

Originally posted by mendhak
And I quote



While I have understood this, I was wondering of some examples of "sets of sets which are not members of themselves."

Off the top of my head, {NULL} would be an example. Is this correct? (How do I type that null symbol, where's NoteMe when you need him? :D )

Can I see more examples of such sets?

A barber shaves everyone in the town that doesn't shave themselves.
Does he shave himself?

btw {{}} doesn't qualify.. i think note's keyboard has that o with / but i don't, so i just type {}

[Acidic]

just wondering, did my example qualify?

[mendhak]

I went over it briefly, but I think Acidic's example qualifies.

If {{}} doesn't qualify, does wossname's example qualify?

{infinity}

I think {infinity} would contain all sets, and therefore include itself.

[mendhak]

Originally posted by kedaman
A barber shaves everyone in the town that doesn't shave themselves.
Does he shave himself?


Aah, you're getting into paradoxes now. I was just writing a compilation of paradoxes (http://www.mendhak.com/thoughts/show.php?id=72), and I did include that one as well. :)



btw {{}} doesn't qualify.. i think note's keyboard has that o with / but i don't, so i just type {}

Can you explain how it doesn't qualify? It contains a ZERO number of sets which are not members of themselves.

[kedaman]

Originally posted by mendhak
I went over it briefly, but I think Acidic's example qualifies.

If {{}} doesn't qualify, does wossname's example qualify?

{infinity}

I think {infinity} would contain all sets, and therefore include itself.
infinity is not a number, so its not even a set

[kedaman]

Originally posted by mendhak
Aah, you're getting into paradoxes now. I was just writing a compilation of paradoxes, and I did include that one as well. :)




Can you explain how it doesn't qualify? It contains a ZERO number of sets which are not members of themselves.
I think you have misunderstood something, it contains a null set, nothing else. To qualify it needs to contain all sets that are not members of themselves. There should only be one such set though, which is the one i mentioned, but this set is a paradox, and thus brings down the whole set theory

[mendhak]

No, I get it now, I wasn't thinking of the difference between 0 and NULL.

Thanks. :afrog:

[wossname]

From Mendhak's site: The sentence "This sentence is false" has no premise and there fore cannot be either true nor false. The question is undefined.




One you might like to add (although its not actually a paradox, but a self-referential statement)...

"is preceded by it's quotation." is preceded by it's quotation.

Also a paradox that is true of all PGA golf courses throughout Britain:

You cannot join a golf club unless you have had your handicap measured properly. You cannot have your handicap measured properly unless you are a member of a golf club.

[kedaman]

wossname: the sentence is a paradox:
G = G = false
=> true = false
=> false

[wossname]

Money makes no cents.

[vbNeo]

Originally posted by kedaman

btw {{}} doesn't qualify.. i think note's keyboard has that o with / but i don't, so i just type {}

You mean: Ø ;)

or maybe Æ and Å ? All members of the scandinavian alphabet - and they're also part of the ascii table(obviously), so just use that instead ...

[mendhak]

Originally posted by wossname
From Mendhak's site: The sentence "This sentence is false" has no premise and there fore cannot be either true nor false. The question is undefined.




One you might like to add (although its not actually a paradox, but a self-referential statement)...

"is preceded by it's quotation." is preceded by it's quotation.

Also a paradox that is true of all PGA golf courses throughout Britain:

You cannot join a golf club unless you have had your handicap measured properly. You cannot have your handicap measured properly unless you are a member of a golf club.

Duly added. :afrog: