set that contains itself seems like the Ouroboros making the last bite. How is that managed? — tim wood
I'm not seeing how you can "without X" and still have any X left - in terms of the notation. — tim wood
No, I am saying there are infinite collections of things that are not a set.
See this link https://math.stackexchange.com/questions/24507/why-did-mathematicians-take-russells-paradox-seriously — EnPassant
The paradox asks the question "Is X a member of itself?"
Let's say Set X = {{a}, {b}, {c},....}
If {X} is a member of X then
Set X = {{a}, {b}, {c},....{X}} — EnPassant
Your notation is confusing. If you want to say that a is a member of X (a ∈ X), you would write that as
X = {a, ...}
which is not the same as
X = {{a}, ...}
{a} is a singleton set with a as the sole member. — SophistiCat
Yes, but X is a set of sets so X = {{a}, {b}, {c},...} but {a, b, c, ...} might be correct too as long as the logic of what I'm saying holds up. — EnPassant
The paradox asks if {X} is a member of X — EnPassant
Set X = {{x}, {y}, {z}}
If X is included
X = {{x}, {y}, {z}, {{x}, {y}, {z}}} — EnPassant
No!
The paradox asks if X is a member of X. — SophistiCat
No, that's not how it works.
X = {{x}, {y}, {z}}
X' = {{x}, {y}, {z}, {{x}, {y}, {z}}}
X ≠ X'
X ∈ X'
X' ∉ X' — SophistiCat
Let 'All sets that do not contain themselves as members' be
a = {x}
b = {y}
c = {z}
d = ... and these sets go on for as long as is necessary, e, f, g, h,... — EnPassant
X ∈ X'
X' ∉ X' — SophistiCat
Those symbols are just Unicode characters that you can copy/paste from anywhere — SophistiCat
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