Set A: Element of Itself? Meaning Explained

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Discussion Overview

The discussion revolves around the concept of whether a set can be an element of itself, exploring implications in set theory, particularly in relation to axioms and paradoxes. Participants examine specific set definitions and their validity within different frameworks of set theory.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions if a set A can be an element of itself, or if it must not be an element of itself, seeking clarification on the implications of such scenarios.
  • Another participant suggests that assuming a set can be an element of itself is valid under certain axioms, referencing non-well-founded set theory.
  • There is a discussion about the sets {x: x=x} and {x: x not an element of x}, with participants noting that the latter does not constitute a set and questioning why it does not represent the null set.
  • One participant asserts that {x: x=x} is classified as a proper class, while another mentions that without the Axiom of Foundation, {x: x not an element of x} might be too large to be a set and cannot be proven to be empty.
  • Participants discuss the implications of the Axiom of Foundation, with one stating that under this axiom, {x: x is not an element of x} is the proper class V, linking it to Russell's paradox.
  • There is a correction from a participant who acknowledges a mental error regarding the classification of {x: x is an element of x} as the empty set with Foundation.

Areas of Agreement / Disagreement

Participants express differing views on the validity of certain sets and axioms, indicating that multiple competing perspectives exist without a clear consensus on the implications of self-containing sets.

Contextual Notes

Limitations include the dependence on specific axioms, such as the Axiom of Foundation, and the unresolved nature of whether certain collections can be classified as sets or proper classes.

ronaldor9
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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.
 

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