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Russell's Paradox:
In a group of people who are shaved there is a single barber. The barber shaves all the people who do not shave themselves. Who shaves the barber?
It can't be the barber himself, because he only shaves people who do not shave themself, thus he can not shave himself. It also can't be anyone else as the barber shaves ALL people who do not shave themselves, so no one else could possibly shave another person.
Therefore it is impossible for all three properties to define a set at the same time:
* everyone is shaved
* The barber shaves people who don't shave themself
* The barber is the only person who can shave other people
This creates two sets one of the people who shave themselves and one shaved by the barber. Since the barber is not allowed to shave himself the sets must be disjoint, but since the barber must be shaved they must also intersect. Since both are not possible there is a contradiction and thus the paradox.
What this means is we can not say any arbitrary set of properties can be used to define a set, as some may give rise to a contradiction.
A more generalized example of the paradox is the idea of a set that contains "all possible sets that do not contain themselves as a set". This is called the universal set. So for example if we had set A defined as the numbers 1, 2 and 3 or A = {1,2,3}, however if set A was A = {A, 1, 2, 3} then it would be excluded. The problem arises when you consider if the universal set includes itself as an element by this definition. If it doesnt contain itself as an element then it qualifies for the rules of inclusion, and thus is included, but by including it it now DOES contain itself as an element thus can not be included. Reaching a paradox once again where the definition of the set as we defined it can not be satisfied. Another example that a set can not simply be defined by an arbitrary collection of properties. Some combinations of properties are invalid while others are not.
@freemo A question oof course is whether the so-called property that is used to define the paradixical Russel set is a property at all or merely nonsense. Attempting to determine whether that set is a member of itself leads to infinite recursion.
I see this as indicating something about sets. We are used to thinking of sets as justbexisting somewhere and we are discovering their properties. But I'm a constructivist, and view our membership conditions as a kind of construction of sets. It's possible to use language to describe all kinds of nonsense, and this one is nonsense.
Not that I mind such nonsense. It's fun. I read fantasy nivels, too, and they're fun.
But trying to build fantasy set theories that can admit such descriptions if fraught with difficulty. You have to avoid plot holes.
@hendrikboom3 well yes, id say a paradox is just a more formal way of saying "nonsense"
@freemo What's interesting, of course, is the *way* it's nonsense. In putting together ZF set theory they found ways to forbid the paradox while still allowing the useful things mathematicians were doing with sets. But there seem to have been multiple attempts to patch it all up, and the one in ZF has gained popularity.
As for what is actually *true* in set theory, I suspect there is no objective way of answering that.
Mathematical philosophers are still working on ways to make logics that allow the Russel set without its paradoxical nature infecting the rest of the set theory.
See for example, Kevin Sharp's note http://kevinscharp.com/ScharpPhilosophyandDefectiveConceptsHandout.pdf
The book he refers to is "Replacing Truth", available for money at Oxford University Press. Or maybe in a library. It may be more detailed than you want to get into.
@freemo Which brings up, I suppose, whether any method of reasoning, any approximate understanding, can ever be considered 'true'. That's a philosophical quagmire I don't want to get into, because I suspect there ix no way out.
@freemo I view truth like a directed set. Lots of different views. But confront different views (assuming they are views and not dogmas) and you can often look at their differences and evidence and come to new understanding that subsumes them. Like we hope quantum mechanics and general relativity will unify. Keep doing that. And in the unattainable (we are finite) limit maybe there's some ideal Truth. But constructively, Truth is a hope, not a reality.
@hendrikboom3 Math isnt really about truth. It is about making useful tools. those tools just are less useful when there are inconsistencies and more useful when everything works according to simple rules without needing long lists of exceptions. I dont see it about reaching an absolute truth, just trying to build a useful tool for analysis.