The set of all sets is proper class, because we can't have all the sets as set. If i have a specific example of infinity collection, how can i understand what is it?
If i have a collection what criteria can i use to decide if it's a set or proper class?
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elementary-set-theory
1 Answers
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The only way to determine whether an infinite collection is a proper class or a set is to (a) prove that it's a set by building it using the axioms of $ZFC$, (b) prove that it's a proper class by using an argument similar to Russell's argument, or (c) prove that it's a proper class by finding a way of putting it in one-to-one correspondence with something you know to be a proper class, like the class of all sets.
For example, the collection of even numbers is a set: $\omega$, the set of all natural numbers, is a set, by the Axiom of Infinity. By the Axiom of Separation, any subclass of $\omega$ that can be described is a set, so the collection of even numbers is a set.
For a more elaborate example, the collection of sets containing exactly two natural numbers is a set: $\omega$ is a set by the Axiom of Infinity. The power set of $\omega$ (the collection of all subsets of $\omega$) is a set, by the Axiom of Power Set. By the Axiom of Separation, we can separate out those subsets of $\omega$ with exactly two elements each.
On the other hand, the collection of all sets with exactly one element each is not a set: if it were, by the Axiom of Replacement we would be able to replace each member with its own element (that is, replace $\{x\}$ with $x$) and get the set of all sets $V$ (since, by the Axiom of Pairing, $\{x\}$ is a set for every $x$).