Set A: Element of Itself? Meaning Explained

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The discussion centers on whether a set A can be an element of itself and the implications of such a scenario. It is noted that assuming this as an axiom allows for the possibility, but without such an assumption, neither outcome can be proven. The conversation also addresses why the sets {x: x=x} and {x: x not an element of x} do not qualify as sets, with the latter being considered a proper class rather than the empty set. The Axiom of Foundation complicates the understanding of these sets, leading to confusion regarding their classification. Ultimately, the conversation highlights the complexities and paradoxes inherent in set theory.
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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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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.
 
ronaldor9 said:
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.)
 
With foundation, {x:x is not an element of x} is the proper class V. In naive set theory it forms the Russel paradox.
 
Oops, I flipped that one mentally to "{x: x is an element of x}" which is the empty set with Foundation.
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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