Set of all sets (Russel’s) paradox is not a paradox because there is no such thing as the set of all sets.

Proof: Let A, B be sets. Then C={A,B,C} is not a set because it is undefined (circular definition).

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- February 25th 2013, 12:17 PMHartlwSet of All Sets (Russel's) Paradox
Set of all sets (Russel’s) paradox is not a paradox because there is no such thing as the set of all sets.

Proof: Let A, B be sets. Then C={A,B,C} is not a set because it is undefined (circular definition). - February 25th 2013, 12:33 PMjakncokeRe: Set of All Sets (Russel's) Paradox
Why can't there exist a set containing all sets? The definition of set under the Cantor definition is a set is a collection of distinct objects.

Russells paradox says if R is a set containing all sets which are not members of themselves. Then R is also included in this set, For R contains sets which are not members of themselves.

At the same time, R is a member of itself.

Thus the paradox. - February 25th 2013, 12:56 PMemakarovRe: Set of All Sets (Russel's) Paradox
- February 25th 2013, 02:11 PMHartlwRe: Set of All Sets (Russel's) Paradox
- February 26th 2013, 01:07 PMHartlwRussel's Paradox
Another point of view. (ϵ’ means does not belong to)

if R = {x│x ϵ’ x } then R ϵ R <-> R ϵ’ R (wiki)

This is not a Paradox because (Theorem) there is no such thing as R ϵ R.

Proof: R_{member}ϵ R_{set}implies R unequal to R. - February 26th 2013, 11:21 PMzzephodRe: Set of All Sets (Russel's) Paradox
- February 27th 2013, 08:09 AMHartlwRe: Set of All Sets (Russel's) Paradox
Axiom: You can’t define something in terms of itself.

If that doesn’t satisfy you, and you accept C={A,C} as a legitimate set, I suggest you substitute the set C={A,C} into the various systems of Axiomatic set theory (which don’t define set) and if you find an inconsistency you may wish to revise that particular axiom. If you don’t find an inconsistency, which I suspect you won’t since the axioms are aware of Russels Paradox, then the only conclusion you can come to is that the axiomatic set has to be revised to reject Russel’s paradox.

Russel published his “Paradox” shortly after Frege published his Foundations of Arithmetic, violating one of Frege’s axioms, which he subsequently revised to account for it. If MHF and this thread were around at the time, he could have saved himself the trouble:

Patrick Suppes Axiomatic Set Theory, pg 5. Dover, 1972

EDIT: If you don't accept the axiom of definition, I can suggest a few books on logic for you to work through. - February 27th 2013, 12:49 PMzzephodRe: Set of All Sets (Russel's) Paradox
- February 27th 2013, 01:19 PMjakncokeRe: Set of All Sets (Russel's) Paradox
If you can't define something in terms of it self, then recursive functions wouldn't be well defined. The natural numbers wouldn't exist. So wouldn't alot of other things in mathematics.

- February 28th 2013, 06:13 AMHartlwRe: Set of All Sets (Russel's) Paradox