# A new point of view on Russell's first paradox

1. Aug 17, 2003

Consider the set M to be "The set of all sets that do not contain themselves as members". Formally: A is an element of M if and only if A is not an element of A. In the sense of Cantor, M is a well-defined set. Does it contain itself? If we assume that it does, it is not a member of M according to the definition. On the other hand, if we assume that M does not contain itself, than it has to be a member of M, again according to the very definition of M. Therefore, the statements "M is a member of M" and "M is not a member of M" both lead to a contradiction. So this must be a contradiction in the underlying theory.

Some example:

if we had an entry on list of all lists which do not contain themselves, then that list must be either incomplete (if it does not list itself) or incorrect (if it does).
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A structural point of view:

Definition A:
( http://www.cut-the-knot.org/selfreference/russell.shtml )
-------------

Sets are defined by the unique properties of their elements
.

One may not mention sets and elements simultaneously, but one notion has no meaning without other.

Let us take as an exapmle, the W set (the set of all positive integers):

{0,1,2,3,...}

By using the empty set (with the Von Neumann Heirarchy), we can show that W has the structure of a set that contain itself as a member of itself:
Code (Text):

0 = { }

1 = {{ }} = {0}
0.
|
|

2 = {{ },{{ }}} = {0,1}
0.    .
|    |
1|____|
|
|

3 = {{ },{{ }},{{ },{{ }}}} = {0,1,2}
0.    .     .    .
|    |     |    |
1|____|     |____|
|          |
2|__________|
|
|

4 = {{ },{{ }},{{ },{{ }}},{{ },{{ }},{{ },{{ }}}}} = {0,1,2,3}
0.    .     .    .      .    .     .    .
|    |     |    |      |    |     |    |
1|____|     |____|      |____|     |____|
|          |           |          |
2|__________|           |__________|
|                      |
3|______________________|
|
|

{0,1,2,3,...}={[B]{ }[/B],[B]{[/B]{ }[B]}[/B],[B]{[/B]{ },{{ }}[B]}[/B],[B]{[/B]{ },{{ }},{{ },{{ }}}[B]}[/B],[B]{[/B]...
|0| |-1-| |----2----| |----------3----------| |--4
|    ^        ^                 ^               ^
|____|        |                 |               |
|           |                 |               |
|___________|                 |               |
|                       |               |
|_______________________|               |
|                           |
|___________________________|

By definition A, the set of all sets that contain themselves as members, must have some kind of the above self structural similarity over scales, by a recursive process.

by a recursive process, I mean that to be a member of yourself is a "never ending story".

Also by definition A, the set of all sets that do not contain themselves as members, must not have this property, therefore the set of all sets that do not contain themselves as members, must not contain itself as a member of itself.

Through this structural point of view, there is no paradox.

This is not logical but a structural point of view on this paradox, and it is based on the simple fact that there can not be any separation between a set and the properties of its contents.

Therefore, the set of all sets that do not contain themselves as members, must have this property, which is:

Not to contain itself as a member of itself.

An example:
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Please look at the structure of a fractal.

It has a self similarity to its content.

The set of all sets with some common property, also has a self similarity with respect to its members, because there cannot be any separation between a set and its members.

We are talking about two kinds of properties:

Property A
------------
Members that contain themselves as members.

Property B
------------
Members that DO NOT contain themselves as members.

Because there can not be any separation between a set and the properties of its members, the set of all sets that do not contain themselves as members is a "property B" set.

The paradox arises when we force "property A" on "property B".

Another Example:
----------------

The honest can't say :"I am a liar" , because he can't lie.

The liar can't say :"I am a liar", because he can't say the truth.

When someone in this boolean universe (where each person can be
honest XOR liar) use the words "All people are liars",
he is included too.

It means that no one in this boolean universe can say those words.

The set of ALL_sets_that_contain_themselves,
must contain itself as a member of itself.

The set of ALL_sets_that_do_not_contain_themselves,
must not contain itself as a member of itself.

Eech set can express only its internal property (the property of its members), therefore no set can contradict itself (the same as the liar paradox) and we have on paradox.

In general, anything that its existence depends on some property, cannot include itself in any ratio with the negation of this property, therefore the question:

" Is the set of ALL_sets_that_do_not_contain_themselves, contain itself or not ?"

is equivalent to:

" is the liar is an honest or not ? " OR " is the honest is a liar or not ?"

All those kinds of questions, are meaningless questions, and they do not lead to any paradox.

What do you think?

Last edited by a moderator: Sep 1, 2003