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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 So after reading that it seems it was little more thn a hard drive platter but the magnetic bit imprinted on it didnt last very long. So it needed to be constantly read and rewritten. Making it a delay line yes, though a very strange one as delay lines go I suppose.
