In ZFC, is the collection $\mathcal{N}$ of nonempty subsets of $\mathbb{Z^{+}}$ a class or set? Please correct me if I am wrong, but so far as I understand the notion of class, a class is a collection of objects which satisfy a certain property $P$. In this example, let $P$ be the property that $A$ is a nonempty subset of $\mathbb{Z^{+}}$. Take $N=\{S|PS \}$. Then is $N$ not just our collection $\mathcal{N}$, and is $N$ not a class?
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1ZFC has the power set axiom. – Angina Seng Oct 17 '18 at 06:14
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Okay, so $\mathcal{N}$ is a set, but is is also a class? – JDivision Oct 17 '18 at 06:28
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1https://math.stackexchange.com/questions/2284709/zfc-with-class-terms-is-any-set-also-a-class (in short: every set is a class, classes which are not sets are called proper classes.) – Asaf Karagila Oct 17 '18 at 06:29
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@AsafKaragila Thank you! +1 – JDivision Oct 17 '18 at 06:32
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I guess we can close this as a duplicate, then. – Asaf Karagila Oct 17 '18 at 06:33