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Two sets A and B are the same, if

$ \displaystyle \forall X: [X \in A \Leftrightarrow X \in B] \land [A \in X \Leftrightarrow B \in X]$

Ehm. I know one explanation that says, "two sets are equal, if they own the same elements". If so, why do we need the second part of the definition? If I read this, I see: "Two sets are equal, if they own the same elements and are elements of some other thing". But then, they are elements of that that is an element of themselves. What are these circles of mind?

Can somebody explain me the sacral meaning of this second part? Thanks.

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    $\begingroup$ I have never seen that definition, but I presume it requires A and B to belong to the same "class". For example you can have an identically named object "Triangle" in a set "Shapes" and also in a set "Strings of characters". $\endgroup$ – David Peterson Jan 5 '17 at 3:12
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    $\begingroup$ What is your source for that definition? $\endgroup$ – bof Jan 5 '17 at 3:17
  • $\begingroup$ @DavidP That is a definition that is given before the Axioms of ZFC. $\endgroup$ – Kirill Jan 5 '17 at 3:25
  • $\begingroup$ @bof, it is difficult for me to say this, as "my" source is the script I got in the university. Maybe, it has some other origin. In words the definition sound like: "Two sets are equal, if they behave identically in regard to the $\in$." Hope the translation was ok :) $\endgroup$ – Kirill Jan 5 '17 at 3:25
  • $\begingroup$ The idea is that "equal" things can be substituted for each other in every fomula: where one appears the other can too, and vice versa. If $=$ is a part of your logical language, this is one of the axioms for the equality symbol, but in ZFC we want to "define" equality (having the same elements, i.e. the axiom extensionality) while keeping the semantics of the logical $=$. $\endgroup$ – Henno Brandsma Jan 5 '17 at 4:56
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The definition of equality both tells us what equality is but also tells us what facts about equal things we may use. So if we are told $ A = B $ then we are told that both that they have the same elements and we are being told that they are elements of the same sets.

When we are dealing at the level of axioms we can't depend on what we expect equality to mean. For example imagine if we had symbols $ \doteq $ and $ \triangleright $. If you had the axiom $ A \doteq B \Leftrightarrow \forall X : [ X \triangleright A \Leftrightarrow X \triangleright B ] $

Then the following graph (with arrows indicating the $ \triangleright $ relationship) would have $ A \doteq B $ but A and B wouldn't act in the way you would expect them to act if they where equal.

enter image description here.

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  • $\begingroup$ 1) And, we can postulate that they are elements of some other set because of the pairing Axiom, right? 2) But: If they have the same set of their elements named C, and, they are elements of some set D - I can easily accept it. But the definition says: some C is the set of elements of A and B, and at the same time A and B are elements of this set C.So, if you spoke about a person A and I spoke about a person B and $x \in y$ means "x is a son of y", we would speak about the same person A=B, if A and B had the same son C, AND, A and B were both sons of C. Is that not "strange"? $\endgroup$ – Kirill Jan 5 '17 at 18:26

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