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
You should consider checking out NBG class-set theory which is an alternative formulation of set theory. — Tommy
My idea is that it can be framed in terms of set theory alone without the invention of classes. — EnPassant
Let Set X = "All sets that do not contain themselves as subsets"\X
I don't see anything wrong with this definition... — EnPassant
The problem with this definition is that the set of all sets that do not contain themselves as subsets is shown, by Russell's Paradox, to be logically contradictory. Your definition requests that we posit an object which is logically contradictory, and then remove
X
X from it. This is akin to requesting the reader to take the smallest prime number with exactly three divisors, subtract it from itself, and then insist that the answer is 0. — Tommy
What's the difference to Russell's Type theory?My argument is to define X without X as a subset of itself regardless of whether it can or can't be such. In this way the paradox is avoided by defining a set that contains 'All sets...' but not X. X is then included in X2 and the paradox is avoided. The same argument is then applied to X2, X3... The result is that the 'Set of all...' is really an infinity of sets each containing the other. — EnPassant
What's the difference to Russell's Type theory? — ssu
Your definition requests that we posit an object which is logically contradictory, and then remove X from it. — Tommy
In this way the paradox is avoided by defining a set that contains 'All sets...' but not X. — EnPassant
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