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Suppose I have the set

$X = \{\varnothing ,\{ \varnothing\} ,\{\{\varnothing\}\} , \{\{\{\varnothing\}\}\} , ..., x_n , \{x_n\}, ... \}$

My book asks which finite and which infinite subsets are models of extensionality? If $Y \subseteq X$ then $Y \vDash$ extensionality iff $Y \cap ((x\setminus y) \cup (y\setminus x)) \ne \varnothing$ for all distinct $x, y \in Y$. It seems to me that all non-empty subsets are models of extensionality, whether they are finite or infinite (except that finite sets must have at least two elements). If $x=x_m$ and $y=x_n$ (the $m$ and $n$ indicate the number of braces around the empty set), then $x_m \setminus x_n = x_{m-n-1} \ne \varnothing$ for $m > n$; and $x_m \setminus x_n = x_m \ne \varnothing$ for $m < n$.

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Note that extensionality holds when $(x\setminus y) \cup (y\setminus x)$ is not empty for $Y$ for $x\ne y$, that means that if $y\ne x$, then there exists some $z\in Y(!)$ such that $z\in x\setminus y$ or $z\in y\setminus x$.

Take $Y=\{\emptyset,\{\{\emptyset\}\}\}$ for example, in this case both $\emptyset$ and $\{\{\emptyset\}\}$ contains no elements in $Y$, so both are empty sets for $Y$, and so if $x=\emptyset,y=\{\{\emptyset\}\}$, we have $\left((x\setminus y) \cup (y\setminus x)\right)^Y$ is empty so extensionality does not hold.

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  • $\begingroup$ And I see that this argument would fail if I had chosen the set of von Neumann natural numbers $X = \{ 0, 1, 2, ... \}$ with $Y=\{ 0, 2 \}$ since $0 \in 2$. Is this why the original $X$ (above) was abandoned as a model for the natural numbers in favor of von Neumann? $\endgroup$ – Robert Singleton Apr 13 '19 at 13:28
  • $\begingroup$ @RobertSingleton one of the reasons, in general transitive sets are extremely nice(like $\in$ well ordering the set) $\endgroup$ – ℋolo Apr 13 '19 at 13:56
  • $\begingroup$ I'm beginning to appreciate transitive sets. In general, it seems that if a subset of $X$ is not transitive, then it will not model extensionality. I'm now trying to prove that the only subsets of $X$ that model extensionality are transitive. $\endgroup$ – Robert Singleton Apr 13 '19 at 14:06
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After thinking about @Holo's reply, I have the full answer to my question (I think):

  • The only finite subsets of $X$ that model extensionality are the sequential subsets $Y=\{x_0, x_1, ..., x_n \}$ where $x_0=\varnothing$ and $x_{m+1}=\{x_m\}$. If an element $x_m$ for $m < n$ is missing from $Y$, then $x_m \notin Y$ but $x_{m+1}\in Y$, in which case $Y \cap (x_{m+1} \setminus x_0) = Y \cap x_{m+1} = \varnothing$ (and $x_0 \setminus x_{m+1}$ is itself empty).

  • The only infinite subset of $X$ that models extensionality is $X$ itself. Again, this is because the elements must be sequential as in the finite argument.

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    $\begingroup$ This answer assumes $x_0=\varnothing\in Y$. Any set of the form $\{x_k,x_{k+1},\dots, x_n\}$ also satisfies extensionality. $\endgroup$ – Alex Kruckman Apr 13 '19 at 18:17
  • $\begingroup$ @AlexKruckman By the same token, any infinite set of the form $\{x_k, x_{k+1}, .... \}$ satisfies extensionality? $\endgroup$ – Robert Singleton Apr 14 '19 at 14:02
  • $\begingroup$ Yes. And now you have all of them. $\endgroup$ – Alex Kruckman Apr 14 '19 at 15:47

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