Set element of itself

  • Thread starter ronaldor9
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92
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Can a set A be an element of A, or can A be not an element of A? And what would such mean in plain-speak?
 
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92
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thanks! By the way, why is it that {x: x=x} and {x: x not an element of x} do not constitute a set? The latter I would think would constitute the null set, but apparently this is wrong.
 

CRGreathouse

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thanks! By the way, why is it that {x: x=x} and {x: x not an element of x} do not constitute a set? The latter I would think would constitute the null set, but apparently this is wrong.
{x: x = x} is a proper class.

I would have thought that, with the Axiom of Foundation, {x: x is not an element of x} would be the empty set. (Without it might be too big to be a set, and can't be proven to be empty.)
 
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With foundation, {x:x is not an element of x} is the proper class V. In naive set theory it forms the Russel paradox.
 

CRGreathouse

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Oops, I flipped that one mentally to "{x: x is an element of x}" which is the empty set with Foundation.
 

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