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I'm going through this set of notes to try to learn a bit more about set theory, because, while I've often encountered papers that use concepts from set theory, I've never actually studied it.

I'm reading over the section on classes, and there are two examples, which are supposedly well known: $\{v_0:(v_0=v_0)\}$ and $\{v_0:(\lnot(v_0=v_0))\}$. The first one seems like it contains everything that is equal to itself, but I'm not entirely sure what the second one is. Is it just the empty set, or some empty class that's similar to the empty set, but is a class instead of a set?

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    $\begingroup$ The first one $\{ v_0:(v_0 = v_0) \}$ is the so-called universal class $V$, because correctly "it contains everything that is equal to itself", and everything is equal to itself. Thus everything satisfy the defining condition... $\endgroup$ – Mauro ALLEGRANZA Oct 15 '14 at 13:12
  • $\begingroup$ Does the second class have a name, or is it just referred to as the empty set? $\endgroup$ – ckersch Oct 15 '14 at 13:21
  • $\begingroup$ Its name is "the empty set" : $\emptyset$. $\endgroup$ – Mauro ALLEGRANZA Oct 15 '14 at 13:22
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The class without elements is in fact a set, the empty set.

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Recasting the definition of your set, it is easy to prove that it is empty.

$\{v_0:(\lnot(v_0=v_0))\} \to \forall a:[a\in X\iff a\ne a]$

Suppose $y\in X$. Then we obtain the contradiction $y\ne y$. Therefore, $\forall y:y\notin X$

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