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#1 2017-04-18 11:05:35

Maharadja
Member
Registered: 2017-01-16
Posts: 3

problems understanding Russell's paradox

I was reading a little bit about sets and of course Russell's paradox was mentioned.

From wiki: "According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's paradox. Symbolically:

"

I do not exactly understand why we have to assume that there is a set R which is the set of all sets. Why do we have to assume that?
I mean, is it legal to start with something like this: "Let R be the biggest natural number. If R is ...".
Should be legal if you wanted to prove that there does (not)exists a biggest natural number but you can not say something about the natural numbers them selves.
Or is the russell paradox to show that such a set R does not exists? Thus that like natural numbers sets are in their own way infinite.

I have a feeling that I'm totally missing the point, so any help would be more than welcome.

btw sorry for my bad english, hope I made myself clear enough.

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#2 2017-04-18 20:23:55

Bob
Administrator
Registered: 2010-06-20
Posts: 10,052

Re: problems understanding Russell's paradox

hi Maharadja

R is a set containing only sets so you might like to start with this:

You can explain the paradox without set notation at all so maybe the following will help.

A librarian makes a series of catalogues like these:

The catalogue of all books with a red cover.
The catalogue of all books written in English.

etc etc.

She notices that some catalogues contain themselves in the listing.  For example: 'the catalogue of all books written in English',  is written in English so it contains itself.  However, the catalogue of all books with a red cover does not have a red cover, so it does not contain itself.

She decides to make a new catalogue; the catalogue of all catalogues that do not contain themselves.  'Books with a red cover' is one entry.

So now we can consider the paradox.

Should 'the catalogue of all catalogues that do not contain themselves', contain itself?  If it does then it shouldn't, and it if doesn't, then it should.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#3 2017-04-19 05:58:21

Maharadja
Member
Registered: 2017-01-16
Posts: 3

Re: problems understanding Russell's paradox

Hi Bob,

Thank you for your response, helped me a lot. I think I understand now with your example how the paradox works.
But I still have some questions if you don’t mind.

We have a library with books.
Then A librarian makes a series of catalogues of the books in the library.
You say he makes them, so I assume they are not part of the library.
Then my question is: Why should 'the catalogue of all books written in English', which is written in English contain itself? I don’t see the logic in this.
The catalogue of all books written in english says only something about the books in the library and nothing about itself.
Yes this catalogue may be a book and may be written in english but who cares, we were only making a catalogue of the  books in the library.

More importantly, how do you know the librarian is a female?

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#4 2017-04-19 06:37:33

Bob
Administrator
Registered: 2010-06-20
Posts: 10,052

Re: problems understanding Russell's paradox

hi Maharadja

OK. I need to put more detail to the story.

A catalogue is a special book; it consists of a list of books of a certain type.  For example, if you are interested in Einstein's theory of relativity you might have a look at 'the catalogue of all books about Einstein's theory'.  Saves time and allows you to make a good choice about which to read first.

But catalogues can be lists of any type of book.  I'm not saying it's such a useful catalogue, but the one that lists all the red books can exist in my imaginary library if I say so.  They are part of the library.

So when we look inside 'the catalogue of all books written in English', one of the entries is this catalogue itself.  It is an example of a catalogue that contains itself.

'The catalogue of all the books in the library' is another self containing catalogue as it is a book and it is in the library.

'The catalogue of all catalogues that contain themselves' is another example.  This catalogue actually lists itself, which means it qualifies as a catalogue that contains itself .............. but, that's ok because it does contain itself.  No contradiction there.

The tricky one is 'the catalogue of all catalogues that don't list themselves'.  Let's take a guess and say it doesn't have itself in the lists.  In that case it satisfies the criterion for inclusion so it should contain itself.  But if it does contain itself then it is one of the ones that don't contain itself.  This is the paradox. dizzy

Here's another way of looking at it.

Fred is a barber and he shaves every man who doesn't shave themselves.  So does he shave himself? 

More importantly, how do you know the librarian is a female?

A good question.  In the UK, it used to be the case that one would say 'he' in giving examples.  But we have had an equality revolution and so I thought I should make an effort not to do this.  So my librarian is a lady.  Fred is a man, so there's a balance here.  You might have spotted a flaw in the barber example.  If so, keep it to yourself; it's only a story anyway.  smile

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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