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Set of all sets (Split from: If I can imagine it, it is possible!)

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On 9/20/2013 at 3:03 PM, ydoaPs said:

So, let's take a look at a specific set: the set of all sets which are not members of themselves. The set of all cats is not a member of the set of all cats-it's a set of cats, not of sets! So, it goes in! Likewise, any set consisting of no sets will go in this set of all sets which are not members of themselves.

 

So, we pose a question: Is this set of all sets which are not members of themselves (from here on out, we'll call it 'R') a member of itself? If R is a member of R, then it fails to meet the requirements to be in R, so it isn't a member of R. That's a contradiction, so that's no good. That means R must not be a member of itself. But what happens if R is a member of itself? If R is a member of itself, it meets the requirement to be in R. Since R is the set of ALL sets meeting this requirements, it goes in. Again we have R both being a member of itself and not being a member of itself. So, either way, we get a contradiction. This means something is logically impossible. But we got this result simply from the definitions of sets and members and from the very conceivable idea that you can group whatever you want together.

 

This is a situation in which something is conceivable, but logically impossible. This means it is not the case that whatever you can imagine is possible. Crackpots, take note: the fact that you can imagine something in no way implies that it is possible. It doesn't matter how clear your perpetual motion device/unified theory/God/electric universe is, imagining it doesn't cut the mustard. This is one of the reasons you NEED the math.

Greetings everyone,

This is my very first post here, and it comes out of pure curiosity. I see that the OP's argument draws from Russel's paradox. 

However, I'm having  a very hard time "imagining" a set of all sets that is not a member of itself.  On the contrary, I can perfectly imagine a set of all cats (like a big balloon filled with cats, for instance).

The challenge for me is that Russel's paradox is an abstract mathematical concept, and I am not able to imagine anything physical out of it.

Quoting the OP, "...from the very conceivable idea that you can group whatever you want together." Again, although grouping things is conceiveable for me, a set of all sets that is not a member of itself is not physically conceivable for me.  The thread title reads, "If I can imagine it, it is a possibility."

Anyone who wishes to help me out here is grandly welcomed!

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!

Moderator Note

This probably deserves a thread of its own. Let us know if it fits better in Mathematics, rather than Philosophy.

 
10 hours ago, D_A said:

However, I'm having  a very hard time "imagining" a set of all sets that is not a member of itself. 

That may be because such a set cannot exist (which is the of the paradox).

Also, a set of abstract concepts (ie. sets) is pretty hard to visualise, anyway. The paradox probably works better when written in mathematical (set theoretical) notation. (I think there is an error in ydoaP's description, as well.)

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A non mathematical example from memory (I remember better doing it this way than copy and paste*):

You have a very well organised library with catalogues which collectively reference all the books. Naturally(?) each catalogue is stored according to its own reference as listed in itself. If a catalogue has no reference to itself in its list it is stored in a random location as it can't be very important.

A new librarian is rather irritated to find that some catalogues don't reference themselves; he doesn't like having to search through the entire library for an obscure catalogue.

Solution: create a catalogue of catalogues which don't include themselves in their list. Keep those catalogues in the locations referenced in this catalogue.

Include a reference in this catalogue to itself so it can be stored in a definite location. Oops, you can't since you're creating a catalogue of catalogues which don't include themselves....

So don't include a reference in this catalogue to itself; then you have to include a reference in this catalogue to itself since it's a catalogue of catalogues which don't include themselves....

So this simple, practical solution to this cataloguing problem is impossible because of Russel's paradox.

 

A couple of things which occurred to me as I was writing:

Perhaps this is related to the problem of finding a catalogue of references to books if you (and the librarian) don't know such a catalogue exists.

Or a more practical problem which often irritates me: I find an interesting old article and it doesn't reference any later articles which include references to it.

 

 

 

* and will learn from SF if my understanding is inadequate.

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14 hours ago, Strange said:
!

Moderator Note

This probably deserves a thread of its own. Let us know if it fits better in Mathematics, rather than Philosophy.

 

That may be because such a set cannot exist (which is the of the paradox).

Also, a set of abstract concepts (ie. sets) is pretty hard to visualise, anyway. The paradox probably works better when written in mathematical (set theoretical) notation. (I think there is an error in ydoaP's description, as well.)

Yes please! I'd like to get the post moved to the math section if possible, as I'd like to challenge the mathematicians among us fairly with this case as well.

7 hours ago, Carrock said:

A non mathematical example from memory (I remember better doing it this way than copy and paste*):

 

* and will learn from SF if my understanding is inadequate.

Thank you for your attempt to help, Carrock. However, I'm still struggling.

You start your post with the words, "A non mathematical example". As soon as you start the argument with this presumption, the rules of the (rigid) set theory are no longer applicable (at least according to my logical thinking).

The librarian is free to include a reference in the catalogue (of catalogues which don't include themselves) to itself, for instance. Or, he/she could just throw the whole lot into the bin out of pure frustration, for that is a tap into the vastness of possibilities that root out of human imagination. In real-life, a human being can come-up with solutions outside of set theory logic for such practical problems.

The gist of my argument is that (I feel) your example example is not a fair real-life implementation of Russel's paradox.

 

P.S.

I didn't understand the last part of your post: "* and will learn from SF if my understanding is inadequate."

 

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1 hour ago, D_A said:

Yes please! I'd like to get the post moved to the math section if possible, as I'd like to challenge the mathematicians among us fairly with this case as well.

!

Moderator Note

Done

 

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3 hours ago, D_A said:

the rules of the (rigid) set theory are no longer applicable (at least according to my logical thinking).

The librarian is free to include a reference in the catalogue (of catalogues which don't include themselves) to itself, for instance.

 

Well yes, but a librarian would want accurate descriptions.

The catalogue title "A catalogue of catalogues which don't include themselves (also including exactly one catalogue which does include itself)" is accurate(ish) but who would want to use it without an explanation of why that one catalogue is included since it obviously isn't quite kosher?

The librarian would soon become an expert on explaining Russell's paradox.

If there were many realistic real-life problems involving Russell's paradox the paradox would have been discovered long before Russell.

 

Finding a different solution here is simple but doesn't negate the inability to create that catalogue of catalogues which don't include themselves.

Quote

The gist of my argument is that (I feel) your example example is not a fair real-life implementation of Russel's paradox.

OK.
 

Quote

 

P.S.

I didn't understand the last part of your post: "* and will learn from SF if my understanding is inadequate."

 

Simply that if I got it wrong a science forums member would likely correct it so that errors wouldn't be perpetuated and I'd learn.

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5 hours ago, D_A said:

P.S.

I didn't understand the last part of your post: "* and will learn from SF if my understanding is inadequate."

We have (nearly all, cranks apart) benefitted in learning something from others here. I know I certainly have.

Cooperation is greater than confrontation.

 

5 hours ago, D_A said:

as I'd like to challenge the mathematicians among us fairly with this case as well.

OK a little easy background first.

It is an very desirable goal in mathematics to be able to use the phrase "for all..." since, if true, it guarantees us an answer to some question we are asking of our mathematical system.

A simple example is the statement For all pairs  a, b of real numbers their sum (a + b ) is another real number, c.

But even with such simple systems we run into trouble when we try to extend this to subtraction, multiplication and division.
In other words the rules of normal arithmetic.

Subtraction and multiplication work out fine, but there is one single exception for division.
We cannot divide by zero.

In this case the answer is easy. We just exclude division by zero and the result is a perfectly workable system.

 

Set theory is much much more complicated.

So Russell discovered the following

If P is a property and f is a function on set S (as then defined in those days)

Then  where x is a member of set S

If P(x) for all x in S  then

[a side note here  -  I am using the convention "If statement then result" is an assertion that given the statement the result is true]


[math]P\left( {f\left( x \right)} \right)\;and\;P\left( S \right)\; \notin \;S[/math]


Now consider the set of all instances of P that is W = {x : P(x)}

It follows that


[math]P\left( {f\left( W \right)} \right)\;and\;P\left( W \right)\; \notin \;W[/math]


But by the definition of W, since P(f(W)) , it follows that   [math]P\left( W \right)\; \in \;W[/math]

which is a contradiction.

 

Full resolution of these paradoxes must deny (or exclude) either the existence of the function f or the set W.

Several methods of avoiding W are in use today.

Firstly in naive set theory we simply ignore it since most practical applications do not encounter such sets.

Secondly Zermelo and others devised a list of axioms restricting the definition of a set to avoid this.

Thirdly  Russell devised a heirarcy of generalities  of set called type theory.

 

 

Edited by studiot

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Would such sets be more tangible if related to computers?

Like pointers which point to a memory address but are used to store a value they represent but are not equal to.

 

I have had an idea of a computer algorithm to sort.

Test a value s, to see if an element of array R.

If s is not an element then set s equal to R.

s is not an element of R but the equivalent of R.

I hope I made sense. But I think there is model that helps picture the paradox.

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