The axiom of extensionality states that if two sets have the same members, then they are identical. In symbols, $(\forall x)(\forall y)((\forall z)(z \in x \leftrightarrow z \in y) \rightarrow x=y)$. But how does one prove, possibly using other axioms of ZFC if necessary, that if two sets belong to the same sets, then they are identical? In symbols, how does one prove that $(\forall x)(\forall y)((\forall z)(x \in z \leftrightarrow y \in z) \rightarrow x=y)$?
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Is that actually true? – Paul Ash Apr 11 '24 at 00:08
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9Let $z = {x}$. If $y \in {x}$, then $y = x$. – Robert Israel Apr 11 '24 at 00:09
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I'm not sure that's true without being more specific. For example, the sets {3.8, 5.9, 7.86} and {2, 4, 9} both belong to the set of real numbers (and rational numbers) yet are not the same set. – Nate Apr 11 '24 at 00:38
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1@Nate Note the $\forall z$ appearing the the proposition in the question. The hypothesis of the proposition asserts that $x$ and $y$ belong to all the same sets; i.e., for any given set $z$, $x \in z$ if and only if $y \in z$. – Gavin Dooley Apr 11 '24 at 02:10
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@RobertIsrael That should be an answer; to the OP, the axiom he's using to form $z$ is $\mathsf{Pairing}$ (with the two operands being the same in this case). – Noah Schweber Apr 11 '24 at 02:12
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Relevant – Mauro ALLEGRANZA Apr 11 '24 at 07:42
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This question (in my opinion) is related to having a consistent set-theoretical model where in such models one wants identical objects (these would be sets of course) of the model to be completely indistinguishable. In a Theory of Sets it is not enough to say that "Two sets having exactly the same elements are identical" more is needed. The notion of being completely indistinguishable.
What is **complete set-theoretical indistinguishability?**
It is easy to construct set-theoretical models where sets having the same elements are not elements of the same sets (i.e., the Axiom of Extensionality fails). Thus, in order to have complete set-theoretical indistinguishability one wants these two conditions
Two sets having exactly the same elements are to be considered as one and the same set.
Two sets having exactly the same elements are to belong to the same sets.
Now any one of the above conditions is taken as the Axiom of Extensionality while the other is taken as motivation for the definition of "$=$".
Using the only basic set-theoretical predicate "$x\ \in\ y$" ($x$ and $y$ are set-theoretical variables) one defines a new predicate "$=$" (set-theoretical identity) in terms of the basic set-theoretical predicate "$\in$" and other logical symbols as follows (for two sets $x$ and $y$)
$$x\ =\ y\ \equiv\ (\forall z)(z\ \in\ x\ \longleftrightarrow\ z\ \in\ y)$$where "$\equiv$" is logical equivalence. Notice that that "$x\ =\ y$" expresses a "property" (or relation) of $x$ and $y$.
Now the Axiom of Extensionality becomes this statement
Identical (or Equal) sets are elements of the same sets.
symbolically $$(\forall x)(\forall y)((\forall z)(x\ \in\ z\ \longleftrightarrow\ y\ \in\ z)\ \longrightarrow\ x\ =\ y)$$So, one does not prove this set-theoretical formula because it is an axiom of the Theory of Sets. Notice that the above formula is what is known as a set-theoretical sentence as opposed to being a property
In most books on Set theory one sees the following set-theoretical formula as the Axiom of Extensionality $$(\forall x)(\forall y)((\forall z)(z\ \in\ x\ \longleftrightarrow\ z\ \in\ y)\ \longrightarrow\ x\ =\ y)$$because condition 1 of complete set-theoretical indistinguishability is taken as the Axiom of Extensionality.
