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Manin gives examples of translating the language of set theory to "argot". The first example is

$\forall x(\neg(x\in\emptyset))$ : "for all (sets) $x$ it is false that $x$ is an element of (the set) $\emptyset$" (or "$\emptyset$ is the empty set").

Well and good. But then he says "The second assertion is only equivalent to the first in the von Neumann universe, where the elements of sets can only be sets, and not real numbers, chairs, or atoms."

How does the equivalence break down? Maybe in some semantics sets of measure zero could be the empty set(s)? But I think I have missed the point of his comment. What are the chairs and atoms supposed to suggest?

John Curran
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    If the formula’s quantifier is about sets, then the formula says that phi has no sets as elements. But if the universe has non-sets in it, then a set with no set-elements is not necessarily empty, as it could contain atoms (l.e. objects which are not themselves sets. In a von Neumann setting, there are no atoms, so the formula does define the empty set since containing no sets is equivalent to containing no elements at all in that setting. – Ned Oct 03 '20 at 16:37
  • Sounds sensible. Thanks. – John Curran Oct 03 '20 at 17:15

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