Considering the notion of a "set of all sets" as a member of itself: Doesn't this really amount to a Banach-Tarski doubling manipulation? In effect, by the inclusion it would seem you are adding an additional set of all sets to the set of all sets. It is like a mathematical set of siamese twins. And where does this additional set come from?
Conjecture - All paradoxes reduce to Banach-Tarski. In set-theoretic terms, it is all boils down to the objectification of an empty set. To paraphrase Hamlet, "To objectify or not to objectify, that is the question." All paradoxes are contradictions that constitute an attempt to reconcile a gestalt - an accounting of an object and the space that it occupies. I.e., A sphere and the space that the sphere occupies. Does the sphere actually "occupy" an equivalent space or does the sphere move in and "push aside" an equivalent amount of space? Does this problem require an assembly of two spheres - one composed of matter and the other of space? There are always two perspectives in the attempt to reconcile the coexsistence of object and space. Positing a set as an element of itself is one viewpoint, Russell's considers the same from another viewpoint in an "anti-set" configuration. Consider Hilbert's Hotel, an infinite set of two-element sbsets comprised of a gestalt pairing of (occupant, hotel room. The problem begins with the introduction of a one element subset from outside the Hilbert set and the paradox is ignited. Hilberts proposition of moving the occupant from room to room does not resolve the paradox, it merely kicks the can down the road. Compare that with the problem of the unification of mathematical physics in an attempt to reconcile spacetime with mass/energy..
Perhaps Pure Mathematics, itself, is the source of all paradox. Every number and every element is infinitely duplicable by convention. There is an infinite supply of each number - I can take any number and multiply it by itself an infinite number of times. Inexaustability is built into the program from the start. That is what makes mathematics universally functional. In the real world, I cannot pull an infinite number of rabbits out of a hat.
There is a barber who shaves today all and only those people who did not shave themselves yesterday. Suppose that the barber shaved himself yesterday. Then the barber does not shave himself today. Suppose that the barber did not shave himself yesterday, then the barber shaves himself today. So the barber shaves himself today if and only if he didn't shave himself yesterday. Contradiction gone with some relative tense differentiation between the clauses. Ditto for Russell's set: There is a set that includes now all and only those sets that did not include themselves previously. Suppose that the set included itself previously. Then the set does not include itself now. Suppose that the set did not include itself previously, etc etc. Think of set inclusion as a mathematical operation like arithmetical addition, performed at a particular time and possibly one of a sequence of such operations performed at different times. When you write AεB you are not making a true or false statement, any more than when you write 2+3, but performing an operation of set inclusion, Similarly for ~(AεB), which can be seen as an act of set exclusion. Two arithmetical operations that fail to make a relative tense differentiation, like 2+3*4 can easily be resolved with the use of brackets, or some other way of making explicit which comes first such as add 2 and 3 then multiply by 4, or multiply 3 by 4 then add 2. The Russell "paradox" is therefore just an order of operations ambiguity.
Considering the notion of a "set of all sets" as a member of itself:
Doesn't this really amount to a Banach-Tarski doubling manipulation? In effect, by the inclusion it would seem you are adding an additional set of all sets to the set of all sets. It is like a mathematical set of siamese twins. And where does this additional set come from?
Conjecture - All paradoxes reduce to Banach-Tarski. In set-theoretic terms, it is all boils down to the objectification of an empty set. To paraphrase Hamlet, "To objectify or not to objectify, that is the question."
All paradoxes are contradictions that constitute an attempt to reconcile a gestalt - an accounting of an object and the space that it occupies. I.e., A sphere and the space that the sphere occupies. Does the sphere actually "occupy" an equivalent space or does the sphere move in and "push aside" an equivalent amount of space? Does this problem require an assembly of two spheres - one composed of matter and the other of space? There are always two perspectives in the attempt to reconcile the coexsistence of object and space. Positing a set as an element of itself is one viewpoint, Russell's considers the same from another viewpoint in an "anti-set" configuration.
Consider Hilbert's Hotel, an infinite set of two-element sbsets comprised of a gestalt pairing of (occupant, hotel room. The problem begins with the introduction of a one element subset from outside the Hilbert set and the paradox is ignited. Hilberts proposition of moving the occupant from room to room does not resolve the paradox, it merely kicks the can down the road.
Compare that with the problem of the unification of mathematical physics in an attempt to reconcile spacetime with mass/energy..
Perhaps Pure Mathematics, itself, is the source of all paradox. Every number and every element is infinitely duplicable by convention. There is an infinite supply of each number - I can take any number and multiply it by itself an infinite number of times.
Inexaustability is built into the program from the start. That is what makes mathematics universally functional. In the real world, I cannot pull an infinite number of rabbits out of a hat.
There is a barber who shaves today all and only those people who did not shave themselves yesterday. Suppose that the barber shaved himself yesterday. Then the barber does not shave himself today. Suppose that the barber did not shave himself yesterday, then the barber shaves himself today. So the barber shaves himself today if and only if he didn't shave himself yesterday.
Contradiction gone with some relative tense differentiation between the clauses. Ditto for Russell's set:
There is a set that includes now all and only those sets that did not include themselves previously. Suppose that the set included itself previously. Then the set does not include itself now. Suppose that the set did not include itself previously, etc etc.
Think of set inclusion as a mathematical operation like arithmetical addition, performed at a particular time and possibly one of a sequence of such operations performed at different times. When you write AεB you are not making a true or false statement, any more than when you write 2+3, but performing an operation of set inclusion, Similarly for ~(AεB), which can be seen as an act of set exclusion.
Two arithmetical operations that fail to make a relative tense differentiation, like 2+3*4 can easily be resolved with the use of brackets, or some other way of making explicit which comes first such as add 2 and 3 then multiply by 4, or multiply 3 by 4 then add 2.
The Russell "paradox" is therefore just an order of operations ambiguity.