Taken partly from an article I just read.

Russels paradox is a problem discovered by Bertrand Russel in 1901 when studying set theory.

He describes a set that contains all sets which do not contain themselves. The set of all pornstars does not contain itself for example, so it can be include in Russel's set, as well as many others.

But what about Russel's set itself? It doesnt contain itself so shouldnt it be included as well? But that would mean it DOES include itself which means it DOESNT belong to the set. And it would keep switching like this, monotonically, forever. Hence the paradox.

If it is monotonic then where we begin in the sequence doesnt matter. So lets do away with that bugbear.

Now if russels set IS the set of all sets that dont contain themselves then we get a paradox.

However if russels set merely *has* as a single subset, all sets that dont contain themselves, then shouldnt russel's set with one level of indirection, contain itself?

  • 4
    In fact shouldnt all sets that DO contain themselves be a subset of sets that DONT, when taken to a sufficient nesting level?

    So russels set would be a subset of sets that contain themselves, taken to 1 level of nesting, and so on..
  • 4
    That's some wise crack right there
  • 4
    Bending the spoon is impossible. You have to understand the inner truth, there is no spoon.
  • 3
    So it's a language paradox, not a logical paradox.
  • 2
    Spotted a Wisecrack from a mile ago ;)
  • 5
    In your formulation, Russell's set contains, as an element, a set that is the set of all sets that don't contain themselves. That's just Russell's paradox again. The enclosed set is impossible to construct since it both can and cannot contain itself. If you accept that impossibility, you can draw any conclusion you want from it, which is clearly inconsistent.

    @Root logic is defined by the language you use to express it. If you want the more mathematical version:

    If we let φ(x) stand for x ∈ x and let R = {x: ~φ(x)}, then R is the set whose members are exactly those objects that are not members of themselves. Contradiction.

    It *is* a logical paradox tied to the ability to use any old property to define membership of a set (unrestricted set comprehension).
  • 2
    Isn't this problem "solved" by Gödel's theorem? There will always be paradoxes in any logical system that contains arithmetics...
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