Set A: Element of Itself? Meaning Explained

In summary, the conversation discusses whether or not set A can be an element of itself, and what this means in plain-speak. It is mentioned that assuming it to be an axiom allows for this, but without assuming it or its negation, it cannot be proven either way. Additionally, the question is raised about why {x: x=x} and {x: x not an element of x} do not constitute a set, with the explanation that the former is a proper class and the latter forms the Russell paradox if assuming the Axiom of Foundation.
  • #1
ronaldor9
92
1
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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  • #3
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.
 
  • #4
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.)
 
  • #5
With foundation, {x:x is not an element of x} is the proper class V. In naive set theory it forms the Russel paradox.
 
  • #6
Oops, I flipped that one mentally to "{x: x is an element of x}" which is the empty set with Foundation.
 

FAQ: Set A: Element of Itself? Meaning Explained

1. What is "Set A: Element of Itself"?

"Set A: Element of Itself" is a concept in set theory where a set contains itself as an element. This means that the set is both a member and a subset of itself.

2. What is the meaning behind "Set A: Element of Itself"?

The meaning behind "Set A: Element of Itself" is a paradoxical concept that challenges traditional set theory. It raises questions about the nature of sets and their relationships to themselves.

3. What are some examples of "Set A: Element of Itself"?

One example is the set of all sets, which contains itself as an element. Another example is the set of all non-self-containing sets, which does not contain itself as an element.

4. How does "Set A: Element of Itself" relate to other mathematical concepts?

"Set A: Element of Itself" is related to other mathematical concepts such as Russell's paradox, which also deals with self-containing sets. It also has implications for concepts like infinity and self-reference.

5. What are the implications of "Set A: Element of Itself" in the scientific community?

The implications of "Set A: Element of Itself" in the scientific community are still being debated. Some argue that it challenges the foundations of mathematics and could potentially lead to new discoveries. Others argue that it is a purely theoretical concept with no practical applications.

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